Limits at infinity worksheet for Calculus

Out at infinity only the highest power of each part matters. Compare the two degrees: the bottom winning gives zero, a tie gives the ratio of the leading coefficients, and the top winning means it runs away. Roots, exponentials and logs work the same way: find the part that grows fastest.

Also taught in Precalculus, at its own level and with its own examples.

Not quite the one you need? Same words, different skill:

Infinite Limits (Calculus). Limits whose answer is a direction, not a number.

Select your difficulty

Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Limits at Infinity Date Period Evaluate each limit, or answer as asked. Find limits of rational functions at infinity $\lim_{x \to \infty} \frac{6}{2 x + 3}$ $\lim_{x \to \infty} \frac{x^{3} + 5 x^{2} - 4}{- 2 x^{3} + 6 x + 6}$ $\lim_{x \to \infty} \frac{- x^{4} - 5 x^{3} - x^{2} + 4 x + 2}{x^{3} + 6 x^{2} + 2}$ $\lim_{x \to \infty} \frac{- 5 x + 3}{2 x + 6}$ $\lim_{x \to \infty} \frac{x^{2} - 3 x + 4}{3 x^{2}}$ Find limits at infinity with roots $\lim_{x \to - \infty} \frac{3 x - 2}{\sqrt{9 x^{2} + 2}}$ $\lim_{x \to \infty} \frac{- x + 7}{\sqrt{25 x^{2} + 3}}$ $\lim_{x \to \infty} \frac{2 x - 2}{\sqrt{x^{2} + 2}}$ $\lim_{x \to \infty} \frac{- 5 x + 9}{\sqrt{x^{2} + 3}}$ $\lim_{x \to \infty} \frac{- 2 x - 2}{\sqrt{9 x^{2} + 5}}$ Find limits at infinity and horizontal asymptotes $\lim_{x \to \infty} \frac{3 e^{x} - 4}{4 e^{x} + 3}$ $\lim_{x \to - \infty} \frac{5 e^{x} - 7}{5 e^{x} + 2}$ Find all horizontal asymptotes of $f (x) = \frac{6 x}{\sqrt{x^{2} + 36}}$ $\lim_{x \to - \infty} \frac{6 e^{x} + 5}{3 e^{x} + 4}$ $\lim_{x \to - \infty} \frac{6 e^{x} - 8}{3 e^{x} + 1}$ Find harder limits at infinity $\lim_{x \to \infty} \frac{6 e^{3 x} - 7}{e^{3 x} - 4}$ $\lim_{x \to - \infty} \frac{2 x + 5}{\sqrt{3 x^{2} - 4}}$ $\lim_{x \to \infty} \frac{- 2 e^{2 x} - 3}{5 e^{2 x} + 2}$ $\lim_{x \to \infty} \frac{\sqrt{3 x^{2} + 6}}{3 x - 6}$ $\lim_{x \to \infty} \frac{6 e^{5 x} + 7}{2 e^{5 x} + 8}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits at Infinity

Date Period

Evaluate each limit, or answer as asked.

Find limits of rational functions at infinity
$\lim_{x \to \infty} \frac{6}{2 x + 3}$
$\lim_{x \to \infty} \frac{x^{3} + 5 x^{2} - 4}{- 2 x^{3} + 6 x + 6}$
$\lim_{x \to \infty} \frac{- x^{4} - 5 x^{3} - x^{2} + 4 x + 2}{x^{3} + 6 x^{2} + 2}$
$\lim_{x \to \infty} \frac{- 5 x + 3}{2 x + 6}$
$\lim_{x \to \infty} \frac{x^{2} - 3 x + 4}{3 x^{2}}$

Find limits at infinity with roots
$\lim_{x \to - \infty} \frac{3 x - 2}{\sqrt{9 x^{2} + 2}}$
$\lim_{x \to \infty} \frac{- x + 7}{\sqrt{25 x^{2} + 3}}$
$\lim_{x \to \infty} \frac{2 x - 2}{\sqrt{x^{2} + 2}}$
$\lim_{x \to \infty} \frac{- 5 x + 9}{\sqrt{x^{2} + 3}}$
$\lim_{x \to \infty} \frac{- 2 x - 2}{\sqrt{9 x^{2} + 5}}$

Find limits at infinity and horizontal asymptotes
$\lim_{x \to \infty} \frac{3 e^{x} - 4}{4 e^{x} + 3}$
$\lim_{x \to - \infty} \frac{5 e^{x} - 7}{5 e^{x} + 2}$
Find all horizontal asymptotes of $f (x) = \frac{6 x}{\sqrt{x^{2} + 36}}$
$\lim_{x \to - \infty} \frac{6 e^{x} + 5}{3 e^{x} + 4}$
$\lim_{x \to - \infty} \frac{6 e^{x} - 8}{3 e^{x} + 1}$

Find harder limits at infinity
$\lim_{x \to \infty} \frac{6 e^{3 x} - 7}{e^{3 x} - 4}$
$\lim_{x \to - \infty} \frac{2 x + 5}{\sqrt{3 x^{2} - 4}}$
$\lim_{x \to \infty} \frac{- 2 e^{2 x} - 3}{5 e^{2 x} + 2}$
$\lim_{x \to \infty} \frac{\sqrt{3 x^{2} + 6}}{3 x - 6}$
$\lim_{x \to \infty} \frac{6 e^{5 x} + 7}{2 e^{5 x} + 8}$



Sigma Prep · sigmaprep.io/worksheets Name Limits at InfinityAnswer keyVersion 1 Date Period Evaluate each limit, or answer as asked. Find limits of rational functions at infinity $\lim_{x \to \infty} \frac{6}{2 x + 3}$ $0$ $\lim_{x \to \infty} \frac{x^{3} + 5 x^{2} - 4}{- 2 x^{3} + 6 x + 6}$ $- \frac{1}{2}$ $\lim_{x \to \infty} \frac{- x^{4} - 5 x^{3} - x^{2} + 4 x + 2}{x^{3} + 6 x^{2} + 2}$ $- \infty$ $\lim_{x \to \infty} \frac{- 5 x + 3}{2 x + 6}$ $- \frac{5}{2}$ $\lim_{x \to \infty} \frac{x^{2} - 3 x + 4}{3 x^{2}}$ $\frac{1}{3}$ Find limits at infinity with roots $\lim_{x \to - \infty} \frac{3 x - 2}{\sqrt{9 x^{2} + 2}}$ $- 1$ $\lim_{x \to \infty} \frac{- x + 7}{\sqrt{25 x^{2} + 3}}$ $- \frac{1}{5}$ $\lim_{x \to \infty} \frac{2 x - 2}{\sqrt{x^{2} + 2}}$ $2$ $\lim_{x \to \infty} \frac{- 5 x + 9}{\sqrt{x^{2} + 3}}$ $- 5$ $\lim_{x \to \infty} \frac{- 2 x - 2}{\sqrt{9 x^{2} + 5}}$ $- \frac{2}{3}$ Find limits at infinity and horizontal asymptotes $\lim_{x \to \infty} \frac{3 e^{x} - 4}{4 e^{x} + 3}$ $\frac{3}{4}$ $\lim_{x \to - \infty} \frac{5 e^{x} - 7}{5 e^{x} + 2}$ $- \frac{7}{2}$ Find all horizontal asymptotes of $f (x) = \frac{6 x}{\sqrt{x^{2} + 36}}$ $y = 6$ and $y = - 6$ $\lim_{x \to - \infty} \frac{6 e^{x} + 5}{3 e^{x} + 4}$ $\frac{5}{4}$ $\lim_{x \to - \infty} \frac{6 e^{x} - 8}{3 e^{x} + 1}$ $- 8$ Find harder limits at infinity $\lim_{x \to \infty} \frac{6 e^{3 x} - 7}{e^{3 x} - 4}$ $6$ $\lim_{x \to - \infty} \frac{2 x + 5}{\sqrt{3 x^{2} - 4}}$ $- \frac{2 \sqrt{3}}{3}$ $\lim_{x \to \infty} \frac{- 2 e^{2 x} - 3}{5 e^{2 x} + 2}$ $- \frac{2}{5}$ $\lim_{x \to \infty} \frac{\sqrt{3 x^{2} + 6}}{3 x - 6}$ $\frac{\sqrt{3}}{3}$ $\lim_{x \to \infty} \frac{6 e^{5 x} + 7}{2 e^{5 x} + 8}$ $3$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits at InfinityAnswer keyVersion 1

Date Period

Evaluate each limit, or answer as asked.

Find limits of rational functions at infinity
$\lim_{x \to \infty} \frac{6}{2 x + 3}$ $0$
$\lim_{x \to \infty} \frac{x^{3} + 5 x^{2} - 4}{- 2 x^{3} + 6 x + 6}$ $- \frac{1}{2}$
$\lim_{x \to \infty} \frac{- x^{4} - 5 x^{3} - x^{2} + 4 x + 2}{x^{3} + 6 x^{2} + 2}$ $- \infty$
$\lim_{x \to \infty} \frac{- 5 x + 3}{2 x + 6}$ $- \frac{5}{2}$
$\lim_{x \to \infty} \frac{x^{2} - 3 x + 4}{3 x^{2}}$ $\frac{1}{3}$

Find limits at infinity with roots
$\lim_{x \to - \infty} \frac{3 x - 2}{\sqrt{9 x^{2} + 2}}$ $- 1$
$\lim_{x \to \infty} \frac{- x + 7}{\sqrt{25 x^{2} + 3}}$ $- \frac{1}{5}$
$\lim_{x \to \infty} \frac{2 x - 2}{\sqrt{x^{2} + 2}}$ $2$
$\lim_{x \to \infty} \frac{- 5 x + 9}{\sqrt{x^{2} + 3}}$ $- 5$
$\lim_{x \to \infty} \frac{- 2 x - 2}{\sqrt{9 x^{2} + 5}}$ $- \frac{2}{3}$

Find limits at infinity and horizontal asymptotes
$\lim_{x \to \infty} \frac{3 e^{x} - 4}{4 e^{x} + 3}$ $\frac{3}{4}$
$\lim_{x \to - \infty} \frac{5 e^{x} - 7}{5 e^{x} + 2}$ $- \frac{7}{2}$
Find all horizontal asymptotes of $f (x) = \frac{6 x}{\sqrt{x^{2} + 36}}$ $y = 6$ and $y = - 6$
$\lim_{x \to - \infty} \frac{6 e^{x} + 5}{3 e^{x} + 4}$ $\frac{5}{4}$
$\lim_{x \to - \infty} \frac{6 e^{x} - 8}{3 e^{x} + 1}$ $- 8$

Find harder limits at infinity
$\lim_{x \to \infty} \frac{6 e^{3 x} - 7}{e^{3 x} - 4}$ $6$
$\lim_{x \to - \infty} \frac{2 x + 5}{\sqrt{3 x^{2} - 4}}$ $- \frac{2 \sqrt{3}}{3}$
$\lim_{x \to \infty} \frac{- 2 e^{2 x} - 3}{5 e^{2 x} + 2}$ $- \frac{2}{5}$
$\lim_{x \to \infty} \frac{\sqrt{3 x^{2} + 6}}{3 x - 6}$ $\frac{\sqrt{3}}{3}$
$\lim_{x \to \infty} \frac{6 e^{5 x} + 7}{2 e^{5 x} + 8}$ $3$

How to do these

$\lim_{x \to \infty} \frac{3 x^{2} + 5}{4 x^{2} - x}$

Both degrees are 2, so it is a tie.
The leading coefficients are 3 and 4.
Everything else is too small to matter.

The answer is $\frac{3}{4}$ .

Where students go wrong

Forgetting that going to minus infinity is not the same question. An odd power changes sign on the left, so a limit that runs up on one side runs down on the other.

Also called limits as x approaches infinity.

What you can put on this worksheet

Limits at infinity: $\lim_{x \to \infty} \frac{x^{2} + 1}{- 4 x^{2} - 3 x + 1}$ → $- \frac{1}{4}$
Roots at infinity: $\lim_{x \to \infty} \frac{4 x}{\sqrt{16 x^{2} + 1}}$ → $1$
Exponentials, bounded functions and both ends: $\lim_{x \to \infty} \frac{4 e^{x} + 1}{3 e^{x} + 6}$ → $\frac{4}{3}$
Polynomials at infinity: $\lim_{x \to - \infty} (4 x^{4} + x^{2} - 5 x + 9)$ → $\infty$
Exponentials, logs, roots that are not squares, oscillation: $\lim_{x \to \infty} \frac{x^{2} + 1}{3 x^{3} - 9}$ → $0$

Questions about these worksheets

Yes. Every limits at infinity sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the limits at infinity questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: limits at infinity, roots at infinity, exponentials, bounded functions and both ends, polynomials at infinity and exponentials, logs, roots that are not squares, oscillation. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting that going to minus infinity is not the same question. An odd power changes sign on the left, so a limit that runs up on one side runs down on the other.

Limits at infinity worksheet for Calculus

Select your difficulty

Find limits of rational functions at infinity

Find limits at infinity with roots

Find limits at infinity and horizontal asymptotes

Find harder limits at infinity

Find limits of rational functions at infinity

Find limits at infinity with roots

Find limits at infinity and horizontal asymptotes

Find harder limits at infinity

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these limits at infinity worksheets free?+

Do the limits at infinity worksheets come with an answer key?+

Are these limits at infinity worksheets printable?+

What grade are these limits at infinity worksheets for?+

Can I get a different version of the same limits at infinity worksheet?+

Is this a free alternative to Kuta Software for limits at infinity?+

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What do students get wrong with limits at infinity?+