AP Calculus AB · Lesson Review · August 31, 2026

Limits at Infinity and End Behavior

Read what a function does far to the left and right, compare dominant powers, and transform indeterminate radical forms into limits you can evaluate.

29-minute edited lesson7 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

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0:0001Limits at Infinity and End Behavior 4:5002A Constant over Infinity 6:5603Equal Degrees 13:5104Larger Numerator Degree 18:3405Larger Denominator Degree 21:0206Exponential End Behavior 23:1907Radical Limits and Conjugates

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02 · Understand

Learning targets

Interpret $\lim_{x\to\infty}f(x)$ and $\lim_{x\to-\infty}f(x)$ as right- and left-end behavior.
Evaluate rational-function limits by comparing numerator and denominator degrees.
Track sign when a rational function grows without bound.
Recognize the end behavior of exponential functions.
Use a conjugate to resolve an $\infty-\infty$ radical form.

03 · Vocabulary

Key definitions

Always evaluate the positive and negative directions separately when the sign or power can change the result.

01Limit at infinity: The value that $f(x)$ approaches as $x$ increases or decreases without bound. It describes end behavior, not substitution of a number called infinity.
02Horizontal asymptote: If $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$ , then $y=L$ is a horizontal asymptote in that direction.
03Degree comparison: For rational functions, compare the highest powers in the numerator and denominator before doing any detailed algebra.
04Unbounded limit: A result such as $+\infty$ or $-\infty$ records unbounded growth and its sign; it is not a finite limit value.
05Conjugate: For a radical difference, multiply by the same terms with the opposite sign so that $(a-b)(a+b)=a^2-b^2$ removes the radicals from the numerator.

04 · Apply

Representative examples

Use structure first: simplify the dominant behavior before reaching for detailed algebra.

Foundation

A constant divided by an unbounded magnitude

$\lim_{x\to\pm\infty}\frac{c}{x}=0$
The numerator stays fixed while the denominator's magnitude grows without bound.
For example, $\lim_{x\to\infty}\frac{4}{x}=0$ .

Equal degrees

Use the ratio of leading coefficients

$\lim_{x\to\infty}\frac{2x^3-5x+1}{7x^3+4x^2}=\frac{2}{7}$
Divide every term by $x^3$ . All terms with a remaining power of $x$ in the denominator approach zero.
Only the leading-coefficient ratio remains.

Degree wins

Compare numerator and denominator powers

If the denominator degree is larger, the quotient approaches $0$ .
If the numerator degree is larger, simplify to the leftover power and track its sign.
$\frac{4x^2+1}{x-3}\sim4x$ gives $+\infty$ as $x\to\infty$ and $-\infty$ as $x\to-\infty$ .

Exponential

Identify which direction decays

$\lim_{x\to-\infty}2^x=0$
$\lim_{x\to\infty}\left(\frac13\right)^x=0$
A base greater than one decays to the left; a base between zero and one decays to the right.

Conjugate

Resolve infinity minus infinity

$\lim_{x\to\infty}\left(\sqrt{9x^2+x}-3x\right)$
Multiply by the conjugate and simplify: $\frac{x}{\sqrt{9x^2+x}+3x}$ .
Factor $x$ from the radical for $x\to\infty$ .
$\lim_{x\to\infty}\frac{1}{\sqrt{9+1/x}+3}=\frac16$ .

05 · Avoid

Common mistakes

Substituting infinity as a number.

Infinity describes unbounded behavior; use limit reasoning instead of ordinary substitution.

Ignoring the negative direction.

Odd powers and other sign-sensitive expressions can behave differently as $x\to-\infty$ .

Reporting only “infinity.”

Distinguish $+\infty$ from $-\infty$ .

Keeping every lower-order term.

For rational functions, the dominant powers reveal the end behavior efficiently.

Using $\sqrt{x^2}=x$ for all real $x$ .

The correct identity is $\sqrt{x^2}=|x|$ .

Stopping at $\infty-\infty$ .

That is an indeterminate form; transform the expression with a conjugate.

06 · Check yourself

Quick check

Try each question before opening its answer.

1What is

\lim_{x\to\infty}\frac{7}{x^2}

$0$ , because the denominator's magnitude grows while the numerator remains constant.

2What controls the limit of a rational function whose numerator and denominator have equal degree?

The ratio of the leading coefficients.

3Why must the two directions be checked for

\frac{4x^2+1}{x-3}

Its dominant behavior is $4x$ , which has opposite signs at positive and negative infinity.

4Which algebraic move helps with

\sqrt{9x^2+x}-3x

Multiply by the conjugate $\sqrt{9x^2+x}+3x$ .

07 · Revisit

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