Infinite limits worksheet for Calculus

When the bottom of a fraction goes to zero and the top does not, the fraction runs off. Which way it runs is the question, and the two sides only agree when the power underneath is even.

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

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

Limits at Infinity (Calculus). What a function settles down to far out.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Infinite Limits Date Period Evaluate each limit. Write ∞ or −∞ for an infinite limit, and DNE for any other limit that does not exist. Find limits at a vertical asymptote $\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$ $\lim_{x \to - {\frac{1}{3}}^{-}} \frac{- 6}{3 x + 1}$ $\lim_{x \to - 3} \frac{- 5}{(x + 3)}$ $\lim_{x \to - {\frac{2}{7}}^{-}} \frac{5}{7 x + 2}$ $\lim_{x \to 5} \frac{- 6}{(x - 5)}$ $\lim_{x \to 3^{+}} \frac{x + 5}{x - 3}$ $\lim_{x \to - 4} \frac{4}{(x + 4)}$ $\lim_{x \to \frac{7}{2}} \frac{6}{{(2 x - 7)}^{2}}$ $\lim_{x \to 2} \frac{1}{(x - 2)}$ $\lim_{x \to 2^{-}} \frac{x + 6}{x - 2}$ $\lim_{x \to - 2} \frac{1}{(x + 2)}$ $\lim_{x \to 2^{-}} \frac{x + 5}{x^{2} - x - 2}$ $\lim_{x \to 4} \frac{4}{(x - 4)}$ $\lim_{x \to 4} \frac{5}{{(x - 4)}^{2}}$ $\lim_{x \to 1} \frac{6}{(x - 1)}$ $\lim_{x \to - 3} \frac{- 3}{{(x + 3)}^{2}}$ $\lim_{x \to - 5} \frac{- 1}{(x + 5)}$ $\lim_{x \to {\frac{3}{2}}^{+}} \frac{3}{2 x - 3}$ $\lim_{x \to - 2} \frac{- 4}{(x + 2)}$ $\lim_{x \to - \frac{2}{3}} \frac{- 6}{{(3 x + 2)}^{2}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Infinite Limits

Date Period

Evaluate each limit. Write ∞ or −∞ for an infinite limit, and DNE for any other limit that does not exist.

Find limits at a vertical asymptote
$\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$
$\lim_{x \to - {\frac{1}{3}}^{-}} \frac{- 6}{3 x + 1}$
$\lim_{x \to - 3} \frac{- 5}{(x + 3)}$
$\lim_{x \to - {\frac{2}{7}}^{-}} \frac{5}{7 x + 2}$
$\lim_{x \to 5} \frac{- 6}{(x - 5)}$
$\lim_{x \to 3^{+}} \frac{x + 5}{x - 3}$
$\lim_{x \to - 4} \frac{4}{(x + 4)}$
$\lim_{x \to \frac{7}{2}} \frac{6}{{(2 x - 7)}^{2}}$
$\lim_{x \to 2} \frac{1}{(x - 2)}$
$\lim_{x \to 2^{-}} \frac{x + 6}{x - 2}$
$\lim_{x \to - 2} \frac{1}{(x + 2)}$
$\lim_{x \to 2^{-}} \frac{x + 5}{x^{2} - x - 2}$
$\lim_{x \to 4} \frac{4}{(x - 4)}$
$\lim_{x \to 4} \frac{5}{{(x - 4)}^{2}}$
$\lim_{x \to 1} \frac{6}{(x - 1)}$
$\lim_{x \to - 3} \frac{- 3}{{(x + 3)}^{2}}$
$\lim_{x \to - 5} \frac{- 1}{(x + 5)}$
$\lim_{x \to {\frac{3}{2}}^{+}} \frac{3}{2 x - 3}$
$\lim_{x \to - 2} \frac{- 4}{(x + 2)}$
$\lim_{x \to - \frac{2}{3}} \frac{- 6}{{(3 x + 2)}^{2}}$



Sigma Prep · sigmaprep.io/worksheets Name Infinite LimitsAnswer keyVersion 1 Date Period Evaluate each limit. Write ∞ or −∞ for an infinite limit, and DNE for any other limit that does not exist. Find limits at a vertical asymptote $\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$ $- \infty$ $\lim_{x \to - {\frac{1}{3}}^{-}} \frac{- 6}{3 x + 1}$ $\infty$ $\lim_{x \to - 3} \frac{- 5}{(x + 3)}$ DNE $\lim_{x \to - {\frac{2}{7}}^{-}} \frac{5}{7 x + 2}$ $- \infty$ $\lim_{x \to 5} \frac{- 6}{(x - 5)}$ DNE $\lim_{x \to 3^{+}} \frac{x + 5}{x - 3}$ $\infty$ $\lim_{x \to - 4} \frac{4}{(x + 4)}$ DNE $\lim_{x \to \frac{7}{2}} \frac{6}{{(2 x - 7)}^{2}}$ $\infty$ $\lim_{x \to 2} \frac{1}{(x - 2)}$ DNE $\lim_{x \to 2^{-}} \frac{x + 6}{x - 2}$ $- \infty$ $\lim_{x \to - 2} \frac{1}{(x + 2)}$ DNE $\lim_{x \to 2^{-}} \frac{x + 5}{x^{2} - x - 2}$ $- \infty$ $\lim_{x \to 4} \frac{4}{(x - 4)}$ DNE $\lim_{x \to 4} \frac{5}{{(x - 4)}^{2}}$ $\infty$ $\lim_{x \to 1} \frac{6}{(x - 1)}$ DNE $\lim_{x \to - 3} \frac{- 3}{{(x + 3)}^{2}}$ $- \infty$ $\lim_{x \to - 5} \frac{- 1}{(x + 5)}$ DNE $\lim_{x \to {\frac{3}{2}}^{+}} \frac{3}{2 x - 3}$ $\infty$ $\lim_{x \to - 2} \frac{- 4}{(x + 2)}$ DNE $\lim_{x \to - \frac{2}{3}} \frac{- 6}{{(3 x + 2)}^{2}}$ $- \infty$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Infinite LimitsAnswer keyVersion 1

Date Period

Evaluate each limit. Write ∞ or −∞ for an infinite limit, and DNE for any other limit that does not exist.

Find limits at a vertical asymptote
$\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$ $- \infty$
$\lim_{x \to - {\frac{1}{3}}^{-}} \frac{- 6}{3 x + 1}$ $\infty$
$\lim_{x \to - 3} \frac{- 5}{(x + 3)}$ DNE
$\lim_{x \to - {\frac{2}{7}}^{-}} \frac{5}{7 x + 2}$ $- \infty$
$\lim_{x \to 5} \frac{- 6}{(x - 5)}$ DNE
$\lim_{x \to 3^{+}} \frac{x + 5}{x - 3}$ $\infty$
$\lim_{x \to - 4} \frac{4}{(x + 4)}$ DNE
$\lim_{x \to \frac{7}{2}} \frac{6}{{(2 x - 7)}^{2}}$ $\infty$
$\lim_{x \to 2} \frac{1}{(x - 2)}$ DNE
$\lim_{x \to 2^{-}} \frac{x + 6}{x - 2}$ $- \infty$
$\lim_{x \to - 2} \frac{1}{(x + 2)}$ DNE
$\lim_{x \to 2^{-}} \frac{x + 5}{x^{2} - x - 2}$ $- \infty$
$\lim_{x \to 4} \frac{4}{(x - 4)}$ DNE
$\lim_{x \to 4} \frac{5}{{(x - 4)}^{2}}$ $\infty$
$\lim_{x \to 1} \frac{6}{(x - 1)}$ DNE
$\lim_{x \to - 3} \frac{- 3}{{(x + 3)}^{2}}$ $- \infty$
$\lim_{x \to - 5} \frac{- 1}{(x + 5)}$ DNE
$\lim_{x \to {\frac{3}{2}}^{+}} \frac{3}{2 x - 3}$ $\infty$
$\lim_{x \to - 2} \frac{- 4}{(x + 2)}$ DNE
$\lim_{x \to - \frac{2}{3}} \frac{- 6}{{(3 x + 2)}^{2}}$ $- \infty$

How to do these

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

Just to the right of 2 the bottom is a tiny positive number.
Three divided by a tiny positive number is huge and positive.
So the limit is infinity.

The answer is $\infty$ .

Where students go wrong

Giving infinity for the two-sided limit when the two sides disagree. Across a simple pole one side runs up and the other runs down, so there is no single answer and the limit does not exist.

Also called limits at essential discontinuities, limits to infinity, vertical asymptote limits or unbounded limits.

What you can put on this worksheet

Limits at a pole: $\lim_{x \to 1^{-}} \frac{6}{(x - 1)}$ → $- \infty$
Non-monic and unfactored bottoms, and trig asymptotes: $\lim_{x \to 5^{+}} \frac{x + 1}{x - 5}$ → $\infty$

Questions about these worksheets

Yes. Every infinite limits 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 infinite limits 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 2: limits at a pole and non-monic and unfactored bottoms, and trig asymptotes. Tick as many as you want and set how many of each, or let it spread them evenly.

Giving infinity for the two-sided limit when the two sides disagree. Across a simple pole one side runs up and the other runs down, so there is no single answer and the limit does not exist.

Infinite limits worksheet for Calculus

Select your difficulty

Find limits at a vertical asymptote

Find limits at a vertical asymptote

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these infinite limits worksheets free?+

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Are these infinite limits worksheets printable?+

What grade are these infinite limits worksheets for?+

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

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

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What do students get wrong with infinite limits?+