Precalculus limits at infinity worksheet

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.

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Sigma Prep · sigmaprep.io/worksheets Name Limits at Infinity Date Period Evaluate each limit. $\lim_{x \to \infty} \frac{6}{2 x + 3}$ $\lim_{x \to \infty} \frac{- x^{3} - 3 x - 1}{- 2 x^{2} - 4}$ $\lim_{x \to \infty} \frac{4 x}{- 5 x + 6}$ $\lim_{x \to \infty} \frac{- 2 x^{3} - 2 x^{2} - 1}{- 4 x^{3} - 5 x^{2} + 5 x}$ $\lim_{x \to \infty} \frac{6 x^{4} + 4 x^{3} - 6 x^{2}}{5 x^{3} - 3 x^{2} + x + 6}$ $\lim_{x \to \infty} \frac{4 x^{2} - 2 x + 6}{2 x^{2} - x + 4}$ $\lim_{x \to \infty} \frac{4 x + 4}{- 5 x + 6}$ $\lim_{x \to \infty} \frac{- x^{2} + 6 x}{- 2 x}$ $\lim_{x \to \infty} \frac{- x^{4} + 2 x - 4}{5 x^{3} - 6 x - 2}$ $\lim_{x \to \infty} \frac{- x}{- 2 x + 2}$ $\lim_{x \to \infty} \frac{- 5 x^{3} - 4}{- 2 x^{2} - 6 x + 1}$ $\lim_{x \to \infty} \frac{- 4 x + 4}{4 x^{2} + 2 x - 5}$ $\lim_{x \to \infty} \frac{6}{- x + 1}$ $\lim_{x \to \infty} \frac{2 x + 6}{- 6 x^{2} - 5 x - 4}$ $\lim_{x \to \infty} \frac{4 x - 1}{2 x + 4}$ $\lim_{x \to \infty} \frac{- 3 x + 5}{5 x}$ $\lim_{x \to \infty} \frac{x^{3} - 5 x^{2} + 3 x}{- x^{3} + x^{2} + 5}$ $\lim_{x \to \infty} \frac{- 6 x^{2} - 5 x}{- 5 x^{3} - 4 x^{2} + 6 x}$ $\lim_{x \to \infty} \frac{- 6}{x + 1}$ $\lim_{x \to \infty} \frac{- x^{3} - 5 x^{2} - x - 6}{2 x^{3} + 6 x^{2} - 4 x + 4}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits at Infinity

Date Period

Evaluate each limit.

$\lim_{x \to \infty} \frac{6}{2 x + 3}$
$\lim_{x \to \infty} \frac{- x^{3} - 3 x - 1}{- 2 x^{2} - 4}$
$\lim_{x \to \infty} \frac{4 x}{- 5 x + 6}$
$\lim_{x \to \infty} \frac{- 2 x^{3} - 2 x^{2} - 1}{- 4 x^{3} - 5 x^{2} + 5 x}$
$\lim_{x \to \infty} \frac{6 x^{4} + 4 x^{3} - 6 x^{2}}{5 x^{3} - 3 x^{2} + x + 6}$
$\lim_{x \to \infty} \frac{4 x^{2} - 2 x + 6}{2 x^{2} - x + 4}$
$\lim_{x \to \infty} \frac{4 x + 4}{- 5 x + 6}$
$\lim_{x \to \infty} \frac{- x^{2} + 6 x}{- 2 x}$
$\lim_{x \to \infty} \frac{- x^{4} + 2 x - 4}{5 x^{3} - 6 x - 2}$
$\lim_{x \to \infty} \frac{- x}{- 2 x + 2}$
$\lim_{x \to \infty} \frac{- 5 x^{3} - 4}{- 2 x^{2} - 6 x + 1}$
$\lim_{x \to \infty} \frac{- 4 x + 4}{4 x^{2} + 2 x - 5}$
$\lim_{x \to \infty} \frac{6}{- x + 1}$
$\lim_{x \to \infty} \frac{2 x + 6}{- 6 x^{2} - 5 x - 4}$
$\lim_{x \to \infty} \frac{4 x - 1}{2 x + 4}$
$\lim_{x \to \infty} \frac{- 3 x + 5}{5 x}$
$\lim_{x \to \infty} \frac{x^{3} - 5 x^{2} + 3 x}{- x^{3} + x^{2} + 5}$
$\lim_{x \to \infty} \frac{- 6 x^{2} - 5 x}{- 5 x^{3} - 4 x^{2} + 6 x}$
$\lim_{x \to \infty} \frac{- 6}{x + 1}$
$\lim_{x \to \infty} \frac{- x^{3} - 5 x^{2} - x - 6}{2 x^{3} + 6 x^{2} - 4 x + 4}$



Sigma Prep · sigmaprep.io/worksheets Name Limits at InfinityAnswer keyVersion 1 Date Period Evaluate each limit. $\lim_{x \to \infty} \frac{6}{2 x + 3}$ $0$ $\lim_{x \to \infty} \frac{- x^{3} - 3 x - 1}{- 2 x^{2} - 4}$ $\infty$ $\lim_{x \to \infty} \frac{4 x}{- 5 x + 6}$ $- \frac{4}{5}$ $\lim_{x \to \infty} \frac{- 2 x^{3} - 2 x^{2} - 1}{- 4 x^{3} - 5 x^{2} + 5 x}$ $\frac{1}{2}$ $\lim_{x \to \infty} \frac{6 x^{4} + 4 x^{3} - 6 x^{2}}{5 x^{3} - 3 x^{2} + x + 6}$ $\infty$ $\lim_{x \to \infty} \frac{4 x^{2} - 2 x + 6}{2 x^{2} - x + 4}$ $2$ $\lim_{x \to \infty} \frac{4 x + 4}{- 5 x + 6}$ $- \frac{4}{5}$ $\lim_{x \to \infty} \frac{- x^{2} + 6 x}{- 2 x}$ $\infty$ $\lim_{x \to \infty} \frac{- x^{4} + 2 x - 4}{5 x^{3} - 6 x - 2}$ $- \infty$ $\lim_{x \to \infty} \frac{- x}{- 2 x + 2}$ $\frac{1}{2}$ $\lim_{x \to \infty} \frac{- 5 x^{3} - 4}{- 2 x^{2} - 6 x + 1}$ $\infty$ $\lim_{x \to \infty} \frac{- 4 x + 4}{4 x^{2} + 2 x - 5}$ $0$ $\lim_{x \to \infty} \frac{6}{- x + 1}$ $0$ $\lim_{x \to \infty} \frac{2 x + 6}{- 6 x^{2} - 5 x - 4}$ $0$ $\lim_{x \to \infty} \frac{4 x - 1}{2 x + 4}$ $2$ $\lim_{x \to \infty} \frac{- 3 x + 5}{5 x}$ $- \frac{3}{5}$ $\lim_{x \to \infty} \frac{x^{3} - 5 x^{2} + 3 x}{- x^{3} + x^{2} + 5}$ $- 1$ $\lim_{x \to \infty} \frac{- 6 x^{2} - 5 x}{- 5 x^{3} - 4 x^{2} + 6 x}$ $0$ $\lim_{x \to \infty} \frac{- 6}{x + 1}$ $0$ $\lim_{x \to \infty} \frac{- x^{3} - 5 x^{2} - x - 6}{2 x^{3} + 6 x^{2} - 4 x + 4}$ $- \frac{1}{2}$ 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.

$\lim_{x \to \infty} \frac{6}{2 x + 3}$ $0$
$\lim_{x \to \infty} \frac{- x^{3} - 3 x - 1}{- 2 x^{2} - 4}$ $\infty$
$\lim_{x \to \infty} \frac{4 x}{- 5 x + 6}$ $- \frac{4}{5}$
$\lim_{x \to \infty} \frac{- 2 x^{3} - 2 x^{2} - 1}{- 4 x^{3} - 5 x^{2} + 5 x}$ $\frac{1}{2}$
$\lim_{x \to \infty} \frac{6 x^{4} + 4 x^{3} - 6 x^{2}}{5 x^{3} - 3 x^{2} + x + 6}$ $\infty$
$\lim_{x \to \infty} \frac{4 x^{2} - 2 x + 6}{2 x^{2} - x + 4}$ $2$
$\lim_{x \to \infty} \frac{4 x + 4}{- 5 x + 6}$ $- \frac{4}{5}$
$\lim_{x \to \infty} \frac{- x^{2} + 6 x}{- 2 x}$ $\infty$
$\lim_{x \to \infty} \frac{- x^{4} + 2 x - 4}{5 x^{3} - 6 x - 2}$ $- \infty$
$\lim_{x \to \infty} \frac{- x}{- 2 x + 2}$ $\frac{1}{2}$
$\lim_{x \to \infty} \frac{- 5 x^{3} - 4}{- 2 x^{2} - 6 x + 1}$ $\infty$
$\lim_{x \to \infty} \frac{- 4 x + 4}{4 x^{2} + 2 x - 5}$ $0$
$\lim_{x \to \infty} \frac{6}{- x + 1}$ $0$
$\lim_{x \to \infty} \frac{2 x + 6}{- 6 x^{2} - 5 x - 4}$ $0$
$\lim_{x \to \infty} \frac{4 x - 1}{2 x + 4}$ $2$
$\lim_{x \to \infty} \frac{- 3 x + 5}{5 x}$ $- \frac{3}{5}$
$\lim_{x \to \infty} \frac{x^{3} - 5 x^{2} + 3 x}{- x^{3} + x^{2} + 5}$ $- 1$
$\lim_{x \to \infty} \frac{- 6 x^{2} - 5 x}{- 5 x^{3} - 4 x^{2} + 6 x}$ $0$
$\lim_{x \to \infty} \frac{- 6}{x + 1}$ $0$
$\lim_{x \to \infty} \frac{- x^{3} - 5 x^{2} - x - 6}{2 x^{3} + 6 x^{2} - 4 x + 4}$ $- \frac{1}{2}$

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 end behavior, horizontal asymptotes or limits as x approaches infinity.

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.

Precalculus is usually taken around 11th or 12th grade, 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.

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.

Precalculus limits at infinity worksheet

Select your difficulty

Limits at Infinity

Limits at InfinityAnswer keyVersion 1

How to do these

Where students go wrong

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