Limits at infinity

Warm-up

Three quick questions from earlier lessons.

Discover

What happens to $\dfrac{1}{x}$ as $x$ grows: $\dfrac{1}{10}, \dfrac{1}{100}, \dfrac{1}{1000}, \dots$ ?

It approaches 0
It has no limit, since it never reaches 0

Key idea

$\displaystyle\lim_{x \to \infty} f(x) = L$ means $f(x)$ gets close to $L$ as $x$ grows; then $y = L$ is a horizontal asymptote.

For a rational function, divide every term by the highest power of

x

in the denominator. Top degree smaller: limit 0. Equal degrees: ratio of the leading coefficients. Top degree larger: no finite limit (

\pm\infty

).

Worked example

Find $\displaystyle\lim_{x \to \infty} \dfrac{6x^2 - x}{2x^2 + 5}$ .

\dfrac{6 - \frac{1}{x}}{2 + \frac{5}{x^2}}

Divide every term by x².

\to \dfrac{6}{2}

1/x and 5/x² go to 0.

The limit is

3

Simplify.

Your turn, guided

Your turn: $\displaystyle\lim_{x \to \infty} \dfrac{4x + 1}{2x - 7}$ .

Ratio of leading coefficients:

\dfrac{4}{2}

Same degree.

The limit is

2

4 ÷ 2.

Practice ★

Find $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^2 + 1}{ 5x^2 + 4}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^2 + 4}{ 2x^2 + 1}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^2 + 4}{ 5x^2 + 4}$ .

Watch out!

Practice ★★

Find $\displaystyle\lim_{x \to \infty} \dfrac{ 6x + 3}{x^2 + 2}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 9x + 6}{x^2 + 6}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 7x + 6}{x^2 + 1}$ .

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. $0$

Exit ticket

Three questions to check you have got it.

Find $\displaystyle\lim_{x \to \infty} \dfrac{ 4x^2 + 1}{ 5x^2 + 4}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^2 + 4}{ 2x^2 + 1}$ .
Find $\displaystyle\lim_{x \to \infty} \dfrac{ 3x^2 + 4}{ 5x^2 + 4}$ .

Summary card

Limits at infinity

Divide by the highest power in the denominator.
Top degree smaller: 0. Equal: ratio of leading coefficients. Larger: $\pm\infty$ .
A finite limit $L$ gives the asymptote $y = L$ .

Watch out: constants do not decide the limit at infinity.

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