Practice sheet

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$f(x) = \dfrac{x^2 - 36}{x - 6}$ is not defined at $x = 6$ . What value does $f(x)$ approach as $x \to 6$ ?
$f(x) = \dfrac{(x - p)(x + 4)}{x - p}$ for $p = 0$ : its graph is the line $y = x + 4$ with a hole. Tap the point the graph approaches as $x \to 0$ .
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 33$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $33$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 21$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $21$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 25$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $25$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 35$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $35$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 22$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $22$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 39$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $39$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 35$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $35$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 24$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $24$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 23$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $23$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 20$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $20$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 23$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $23$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 20$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $20$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 27$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $27$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 31$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $31$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 32$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $32$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 28$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $28$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 34$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $34$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 21$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $21$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 28$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $28$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 26$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $26$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 30$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $30$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 20$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $20$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 24$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $24$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 33$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $33$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 25$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $25$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 31$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $31$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 27$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $27$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 40$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $40$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 22$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $22$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 40$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $40$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 23$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $23$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 31$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $31$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 24$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $24$
3. It does not exist
$f(x) = x^2$ for $x \ne 5$ , and $f(5) = 37$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. $25$
2. $37$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 36$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $36$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 21$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $21$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 37$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $37$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 34$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $34$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 27$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $27$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 26$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $26$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 29$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $29$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 32$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $32$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 37$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $37$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 39$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $39$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 38$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $38$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 33$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $33$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 26$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $26$
3. It does not exist
$f(x) = x^2$ for $x \ne 3$ , and $f(3) = 38$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. $9$
2. $38$
3. It does not exist
$f(x) = x^2$ for $x \ne 4$ , and $f(4) = 25$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. $16$
2. $25$
3. It does not exist
$f(x) = x^2$ for $x \ne 2$ , and $f(2) = 25$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. $4$
2. $25$
3. It does not exist
Find $\displaystyle\lim_{x \to 7} \dfrac{x^2 - 15x + 56}{x - 7}$ .
Find $\displaystyle\lim_{h \to 0} \dfrac{(8 + h)^2 - 64}{h}$ .
Find $\displaystyle\lim_{x \to 9} \dfrac{\sqrt{x} - 3}{x - 9}$ .
$f(x) = \begin{cases} x + 6 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $10$
$f(x) = \begin{cases} x + 7 & \text{if } x < 3 \\ 4x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $12$
$f(x) = \begin{cases} x + 2 & \text{if } x < 5 \\ 4x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $20$
$f(x) = \begin{cases} x + 9 & \text{if } x < 1 \\ 2x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $2$
$f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ 4x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $3$
3. $8$
$f(x) = \begin{cases} x + 9 & \text{if } x < 6 \\ 4x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $15$
3. $24$
$f(x) = \begin{cases} x + 2 & \text{if } x < 2 \\ 5x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $10$
$f(x) = \begin{cases} x + 9 & \text{if } x < 6 \\ 5x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $15$
3. $30$
$f(x) = \begin{cases} x + 6 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $6$
$f(x) = \begin{cases} x + 2 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $18$
$f(x) = \begin{cases} x + 7 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $18$
$f(x) = \begin{cases} x + 1 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $5$
3. $12$
$f(x) = \begin{cases} x + 1 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $6$
$f(x) = \begin{cases} x + 9 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $12$
$f(x) = \begin{cases} x + 5 & \text{if } x < 4 \\ 2x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $8$
$f(x) = \begin{cases} x + 7 & \text{if } x < 2 \\ 5x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $10$
$f(x) = \begin{cases} x + 1 & \text{if } x < 6 \\ 4x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $24$
$f(x) = \begin{cases} x + 7 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $12$
$f(x) = \begin{cases} x + 4 & \text{if } x < 6 \\ 5x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $30$
$f(x) = \begin{cases} x + 4 & \text{if } x < 1 \\ 3x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $5$
3. $3$
$f(x) = \begin{cases} x + 1 & \text{if } x < 3 \\ 3x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $9$
$f(x) = \begin{cases} x + 9 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $12$
3. $6$
$f(x) = \begin{cases} x + 7 & \text{if } x < 2 \\ 3x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $6$
$f(x) = \begin{cases} x + 4 & \text{if } x < 6 \\ 2x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $12$
$f(x) = \begin{cases} x + 8 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $14$
3. $18$
$f(x) = \begin{cases} x + 9 & \text{if } x < 4 \\ 2x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $8$
$f(x) = \begin{cases} x + 6 & \text{if } x < 5 \\ 4x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $20$
$f(x) = \begin{cases} x + 7 & \text{if } x < 2 \\ 4x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $8$
$f(x) = \begin{cases} x + 2 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $6$
3. $12$
$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 3x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $2$
3. $3$
$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 2x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $3$
3. $2$
$f(x) = \begin{cases} x + 2 & \text{if } x < 2 \\ 3x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $6$
$f(x) = \begin{cases} x + 7 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $5$
$f(x) = \begin{cases} x + 9 & \text{if } x < 3 \\ 5x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $12$
3. $15$
$f(x) = \begin{cases} x + 3 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $5$
$f(x) = \begin{cases} x + 8 & \text{if } x < 2 \\ 2x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $4$
$f(x) = \begin{cases} x + 7 & \text{if } x < 4 \\ 5x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $20$
$f(x) = \begin{cases} x + 8 & \text{if } x < 5 \\ 5x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $25$
$f(x) = \begin{cases} x + 4 & \text{if } x < 1 \\ 2x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $5$
3. $2$
$f(x) = \begin{cases} x + 9 & \text{if } x < 1 \\ 4x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $4$
$f(x) = \begin{cases} x + 5 & \text{if } x < 6 \\ 2x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $12$
$f(x) = \begin{cases} x + 7 & \text{if } x < 1 \\ 4x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $4$
$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $6$
$f(x) = \begin{cases} x + 5 & \text{if } x < 2 \\ 5x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $10$
$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $15$
$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 5x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $25$
$f(x) = \begin{cases} x + 1 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $6$
3. $10$
$f(x) = \begin{cases} x + 7 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $12$
3. $15$
$f(x) = \begin{cases} x + 2 & \text{if } x < 4 \\ 4x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $6$
3. $16$
$f(x) = \begin{cases} x + 9 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $15$
3. $18$
Find $\displaystyle\lim_{x \to 0^-} \dfrac{|3x|}{x}$ .
Find $\displaystyle\lim_{x \to 0} \dfrac{\sin(6x)}{x}$ .
Find $\displaystyle\lim_{x \to 0} \dfrac{\sin(3x)}{\sin(9x)}$ .
What is $\displaystyle\lim_{x \to 0} x^{ 2 }\sin\left(\dfrac{1}{x}\right)$ ?
1. 0, by the squeeze theorem
2. It does not exist, because $\sin(1/x)$ oscillates
3. 1
What is $\displaystyle\lim_{x \to 0} x^{ 3 }\sin\left(\dfrac{1}{x}\right)$ ?
1. 0, by the squeeze theorem
2. It does not exist, because $\sin(1/x)$ oscillates
3. 1
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^2 + 1}{ 5x^2 + 4}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 6x + 3}{x^2 + 2}$ .
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 5}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 2x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $2$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 9}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^3}{x^2 + 7}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $4$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 9}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 8}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 3}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $4$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 8x^3}{x^2 + 1}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $8$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 7x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $7$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 5}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 8x^3}{x^2 + 5}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $8$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 2x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $2$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 2x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $2$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $4$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $4$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 1}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 8}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 1}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 2x^3}{x^2 + 9}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $2$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 8}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 2x^3}{x^2 + 1}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $2$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 7}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 8x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $8$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 7}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 8x^3}{x^2 + 4}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $8$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 7x^3}{x^2 + 8}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $7$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 7}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^3}{x^2 + 6}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $3$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 5x^3}{x^2 + 7}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $5$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 1}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 7x^3}{x^2 + 2}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $7$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 3}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 8x^3}{x^2 + 8}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $8$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 9}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 7x^3}{x^2 + 3}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $7$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 2x^3}{x^2 + 7}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $2$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 6x^3}{x^2 + 9}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $6$
3. $0$
What is $\displaystyle\lim_{x \to \infty} \dfrac{ 9x^3}{x^2 + 3}$ ?
1. It is $\infty$ : the fraction grows without bound
2. $9$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 7 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^+} \dfrac{ 4 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^+} \dfrac{ 6 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^+} \dfrac{ 1 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^+} \dfrac{ 7 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^+} \dfrac{ 1 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^+} \dfrac{ 3 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^+} \dfrac{ 2 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 5 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 9 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^+} \dfrac{ 5 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 8 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^+} \dfrac{ 8 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 2 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^+} \dfrac{ 7 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^+} \dfrac{ 9 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^+} \dfrac{ 2 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^+} \dfrac{ 9 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 3 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^+} \dfrac{ 8 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 4 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^+} \dfrac{ 8 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^+} \dfrac{ 8 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^+} \dfrac{ 4 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^+} \dfrac{ 2 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 2 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 4 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^+} \dfrac{ 1 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 6 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 5 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^+} \dfrac{ 3 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 1 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^+} \dfrac{ 9 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 6 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^+} \dfrac{ 4 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 8 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^+} \dfrac{ 4 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^+} \dfrac{ 6 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^+} \dfrac{ 2 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^+} \dfrac{ 4 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 3 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^+} \dfrac{ 6 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^+} \dfrac{ 2 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^+} \dfrac{ 1 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^+} \dfrac{ 4 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^+} \dfrac{ 5 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^+} \dfrac{ 5 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^+} \dfrac{ 2 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^+} \dfrac{ 7 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^+} \dfrac{ 8 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^-} \dfrac{ 6 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^-} \dfrac{ 7 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^-} \dfrac{ 9 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^-} \dfrac{ 4 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 7 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 5 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^-} \dfrac{ 5 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 8 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^-} \dfrac{ 2 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^-} \dfrac{ 5 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^-} \dfrac{ 1 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^-} \dfrac{ 9 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 6 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 4 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 2 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^-} \dfrac{ 2 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^-} \dfrac{ 9 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^-} \dfrac{ 8 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^-} \dfrac{ 3 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^-} \dfrac{ 9 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 2 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 3 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^-} \dfrac{ 8 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^-} \dfrac{ 2 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^-} \dfrac{ 1 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 4 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 5 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^-} \dfrac{ 2 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 8 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^-} \dfrac{ 6 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^-} \dfrac{ 7 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^-} \dfrac{ 9 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^-} \dfrac{ 3 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 5^-} \dfrac{ 4 }{x - 5}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 2^-} \dfrac{ 8 }{x - 2}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^-} \dfrac{ 1 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^-} \dfrac{ 1 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^-} \dfrac{ 1 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^-} \dfrac{ 4 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 6^-} \dfrac{ 7 }{x - 6}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^-} \dfrac{ 5 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 9 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 8^-} \dfrac{ 9 }{x - 8}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 3^-} \dfrac{ 3 }{x - 3}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^-} \dfrac{ 8 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 7 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 1^-} \dfrac{ 5 }{x - 1}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 4^-} \dfrac{ 1 }{x - 4}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 9^-} \dfrac{ 4 }{x - 9}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
What is $\displaystyle\lim_{x \to 7^-} \dfrac{ 3 }{x - 7}$ ?
1. $+\infty$
2. $-\infty$
3. $0$
$f(x) = \dfrac{x + 8}{x^2 - 9}$ has vertical asymptotes. What is the positive one, $x = c$ ?
For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 4x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?
Is $f(x) = \dfrac{x^2 - 25}{x - 5}$ continuous at $x = 5$ ?
1. No: $f(5)$ is not defined
2. Yes: the limit exists, it is $10$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 81}{x - 9}$ continuous at $x = 9$ ?
1. No: $f(9)$ is not defined
2. Yes: the limit exists, it is $18$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 4}{x - 2}$ continuous at $x = 2$ ?
1. No: $f(2)$ is not defined
2. Yes: the limit exists, it is $4$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 49}{x - 7}$ continuous at $x = 7$ ?
1. No: $f(7)$ is not defined
2. Yes: the limit exists, it is $14$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 64}{x - 8}$ continuous at $x = 8$ ?
1. No: $f(8)$ is not defined
2. Yes: the limit exists, it is $16$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 36}{x - 6}$ continuous at $x = 6$ ?
1. No: $f(6)$ is not defined
2. Yes: the limit exists, it is $12$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 1}{x - 1}$ continuous at $x = 1$ ?
1. No: $f(1)$ is not defined
2. Yes: the limit exists, it is $2$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 16}{x - 4}$ continuous at $x = 4$ ?
1. No: $f(4)$ is not defined
2. Yes: the limit exists, it is $8$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 9}{x - 3}$ continuous at $x = 3$ ?
1. No: $f(3)$ is not defined
2. Yes: the limit exists, it is $6$
3. No: the limit does not exist
$f$ is continuous on $[1, 9]$ , with $f(1) = -1$ and $f(9) = 7$ . Must $f$ have a zero between 1 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 5]$ , with $f(3) = -2$ and $f(5) = 2$ . Must $f$ have a zero between 3 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 7]$ , with $f(3) = -7$ and $f(7) = 2$ . Must $f$ have a zero between 3 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -1$ and $f(5) = 4$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 8]$ , with $f(3) = -7$ and $f(8) = 1$ . Must $f$ have a zero between 3 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 7]$ , with $f(1) = -9$ and $f(7) = 9$ . Must $f$ have a zero between 1 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 6]$ , with $f(3) = -8$ and $f(6) = 3$ . Must $f$ have a zero between 3 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 5]$ , with $f(2) = -8$ and $f(5) = 7$ . Must $f$ have a zero between 2 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -9$ and $f(5) = 6$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 9]$ , with $f(0) = -4$ and $f(9) = 8$ . Must $f$ have a zero between 0 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 8]$ , with $f(0) = -1$ and $f(8) = 8$ . Must $f$ have a zero between 0 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 5]$ , with $f(0) = -2$ and $f(5) = 6$ . Must $f$ have a zero between 0 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 6]$ , with $f(3) = -9$ and $f(6) = 1$ . Must $f$ have a zero between 3 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 7]$ , with $f(1) = -3$ and $f(7) = 4$ . Must $f$ have a zero between 1 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 6]$ , with $f(2) = -2$ and $f(6) = 3$ . Must $f$ have a zero between 2 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -6$ and $f(5) = 8$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -9$ and $f(5) = 1$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 9]$ , with $f(3) = -8$ and $f(9) = 8$ . Must $f$ have a zero between 3 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 6]$ , with $f(0) = -2$ and $f(6) = 7$ . Must $f$ have a zero between 0 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 9]$ , with $f(1) = -9$ and $f(9) = 6$ . Must $f$ have a zero between 1 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 6]$ , with $f(2) = -7$ and $f(6) = 7$ . Must $f$ have a zero between 2 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 7]$ , with $f(3) = -9$ and $f(7) = 6$ . Must $f$ have a zero between 3 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 5]$ , with $f(0) = -4$ and $f(5) = 1$ . Must $f$ have a zero between 0 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 7]$ , with $f(3) = -9$ and $f(7) = 8$ . Must $f$ have a zero between 3 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 8]$ , with $f(1) = -8$ and $f(8) = 1$ . Must $f$ have a zero between 1 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 5]$ , with $f(3) = -7$ and $f(5) = 1$ . Must $f$ have a zero between 3 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 9]$ , with $f(1) = -2$ and $f(9) = 7$ . Must $f$ have a zero between 1 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 7]$ , with $f(3) = -7$ and $f(7) = 5$ . Must $f$ have a zero between 3 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -2$ and $f(5) = 2$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -2$ and $f(5) = 3$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 7]$ , with $f(3) = -4$ and $f(7) = 3$ . Must $f$ have a zero between 3 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -2$ and $f(5) = 6$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 7]$ , with $f(2) = -1$ and $f(7) = 1$ . Must $f$ have a zero between 2 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -1$ and $f(5) = 5$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 9]$ , with $f(2) = -3$ and $f(9) = 5$ . Must $f$ have a zero between 2 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 8]$ , with $f(2) = -8$ and $f(8) = 3$ . Must $f$ have a zero between 2 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 8]$ , with $f(1) = -6$ and $f(8) = 5$ . Must $f$ have a zero between 1 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 5]$ , with $f(1) = -6$ and $f(5) = 6$ . Must $f$ have a zero between 1 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 6]$ , with $f(2) = -8$ and $f(6) = 5$ . Must $f$ have a zero between 2 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 8]$ , with $f(2) = -7$ and $f(8) = 1$ . Must $f$ have a zero between 2 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 6]$ , with $f(2) = -2$ and $f(6) = 2$ . Must $f$ have a zero between 2 and 6?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[2, 9]$ , with $f(2) = -5$ and $f(9) = 2$ . Must $f$ have a zero between 2 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 5]$ , with $f(0) = -2$ and $f(5) = 4$ . Must $f$ have a zero between 0 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 8]$ , with $f(3) = -4$ and $f(8) = 6$ . Must $f$ have a zero between 3 and 8?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[0, 7]$ , with $f(0) = -4$ and $f(7) = 6$ . Must $f$ have a zero between 0 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 5]$ , with $f(3) = -6$ and $f(5) = 4$ . Must $f$ have a zero between 3 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[1, 9]$ , with $f(1) = -7$ and $f(9) = 9$ . Must $f$ have a zero between 1 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 9]$ , with $f(3) = -5$ and $f(9) = 6$ . Must $f$ have a zero between 3 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 5]$ , with $f(3) = -5$ and $f(5) = 4$ . Must $f$ have a zero between 3 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 7]$ , with $f(3) = -1$ and $f(7) = 7$ . Must $f$ have a zero between 3 and 7?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f(x) = \dfrac{1}{x}$ has $f(-4) < 0$ and $f(4) > 0$ . Does $f$ have a zero in $(-4, 4)$ ?
1. No: $f$ is not continuous on $[-4, 4]$ , so the theorem does not apply
2. Yes, by the Intermediate Value Theorem
3. Yes, at $x = 0$
$f(x) = \dfrac{1}{x}$ has $f(-1) < 0$ and $f(1) > 0$ . Does $f$ have a zero in $(-1, 1)$ ?
1. No: $f$ is not continuous on $[-1, 1]$ , so the theorem does not apply
2. Yes, by the Intermediate Value Theorem
3. Yes, at $x = 0$
$f(x) = \dfrac{1}{x}$ has $f(-2) < 0$ and $f(2) > 0$ . Does $f$ have a zero in $(-2, 2)$ ?
1. No: $f$ is not continuous on $[-2, 2]$ , so the theorem does not apply
2. Yes, by the Intermediate Value Theorem
3. Yes, at $x = 0$
$f(x) = \dfrac{1}{x}$ has $f(-3) < 0$ and $f(3) > 0$ . Does $f$ have a zero in $(-3, 3)$ ?
1. No: $f$ is not continuous on $[-3, 3]$ , so the theorem does not apply
2. Yes, by the Intermediate Value Theorem
3. Yes, at $x = 0$
$f(x) = \dfrac{1}{x}$ has $f(-5) < 0$ and $f(5) > 0$ . Does $f$ have a zero in $(-5, 5)$ ?
1. No: $f$ is not continuous on $[-5, 5]$ , so the theorem does not apply
2. Yes, by the Intermediate Value Theorem
3. Yes, at $x = 0$
$f(x) = x^3 + x - 3$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .

Answers

$12$
$(0, 4)$
$16$
$-1$
$16$
$\frac{1}{6}$
It does not exist: the left and right limits differ
$-3$
$6$
$\frac{1}{3}$
0, by the squeeze theorem
$\frac{4}{5}$
$0$
It is $\infty$ : the fraction grows without bound
$+\infty$
$-\infty$
$3$
$3$
No: $f(5)$ is not defined
Yes, by the Intermediate Value Theorem
No: $f$ is not continuous on $[-4, 4]$ , so the theorem does not apply
$1$