One-sided limits

Warm-up

Three quick questions from earlier lessons.

Discover

A parking garage charges 5 dollars for up to 1 hour and 9 dollars for more. As the time approaches 1 hour from below, the price is 5; from above, it is 9. Does the price approach one single number near 1 hour?

No: 5 from the left, 9 from the right
Yes: 5

Key idea

We can approach $a$ from one side only.

\displaystyle\lim_{x \to a^-} f(x)

uses

x < a

;

\displaystyle\lim_{x \to a^+} f(x)

uses

x > a

. The two-sided limit exists exactly when both one-sided limits exist and are equal.

Worked example

$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 3 - x & \text{if } x \ge 1 \end{cases}$ . Find $\displaystyle\lim_{x \to 1} f(x)$ .

\displaystyle\lim_{x \to 1^-} f(x) = 1

x < 1: use x².

\displaystyle\lim_{x \to 1^+} f(x) = 2

x > 1: use 3 − x.

The limit does not exist

1 ≠ 2.

Your turn, guided

Your turn: same idea with $g(x) = \begin{cases} x + 4 & \text{if } x < 2 \\ 3x & \text{if } x \ge 2 \end{cases}$ .

\displaystyle\lim_{x \to 2^-} g(x) = 6

Left: 2 + 4.

\displaystyle\lim_{x \to 2^+} g(x) = 6

, so the limit exists and is 6

Right: 3 · 2.

Practice ★

$f(x) = \begin{cases} x + 3 & \text{if } x < 2 \\ 2x & \text{if } x \ge 2 \end{cases}$
$f(x) = \begin{cases} x + 7 & \text{if } x < 4 \\ 2x & \text{if } x \ge 4 \end{cases}$
$f(x) = \begin{cases} x + 1 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$

Watch out!

Practice ★★

$f(x) = \begin{cases} x + 6 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . 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. $12$

Find $\displaystyle\lim_{x \to 0^-} \dfrac{|3x|}{x}$ .
Find $\displaystyle\lim_{x \to 0^-} \dfrac{|5x|}{x}$ .
Find $\displaystyle\lim_{x \to 0^-} \dfrac{|8x|}{x}$ .

Exit ticket

Three questions to check you have got it.

Find $\displaystyle\lim_{x \to 0^-} \dfrac{|3x|}{x}$ .
Find $\displaystyle\lim_{x \to 0^-} \dfrac{|5x|}{x}$ .
Find $\displaystyle\lim_{x \to 0^-} \dfrac{|8x|}{x}$ .

Summary card

One-sided limits

$x \to a^-$ : from the left; $x \to a^+$ : from the right.
The limit exists only if both sides agree.

Watch out: one side alone is not the two-sided limit.

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