Infinite limits worksheet

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.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Infinite Limits Date Period Evaluate each limit. Write DNE if the limit does not exist. $\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$ $\lim_{x \to - 3} \frac{6}{(x + 3)}$ $\lim_{x \to - 1^{-}} \frac{- 2}{(x + 1)}$ $\lim_{x \to - 1^{-}} \frac{- 4}{{(x + 1)}^{2}}$ $\lim_{x \to - 5^{-}} \frac{- 5}{{(x + 5)}^{2}}$ $\lim_{x \to 2} \frac{7}{(x - 2)}$ $\lim_{x \to - 1} \frac{8}{(x + 1)}$ $\lim_{x \to - 4^{+}} \frac{- 4}{(x + 4)}$ $\lim_{x \to 1} \frac{- 3}{(x - 1)}$ $\lim_{x \to 4^{-}} \frac{- 8}{{(x - 4)}^{2}}$ $\lim_{x \to 2} \frac{8}{{(x - 2)}^{2}}$ $\lim_{x \to 4} \frac{1}{(x - 4)}$ $\lim_{x \to - 3} \frac{5}{{(x + 3)}^{2}}$ $\lim_{x \to - 4} \frac{8}{(x + 4)}$ $\lim_{x \to - 4^{+}} \frac{- 1}{(x + 4)}$ $\lim_{x \to 1} \frac{1}{{(x - 1)}^{2}}$ $\lim_{x \to - 3^{-}} \frac{- 3}{{(x + 3)}^{2}}$ $\lim_{x \to 2^{-}} \frac{3}{(x - 2)}$ $\lim_{x \to - 2^{+}} \frac{6}{(x + 2)}$ $\lim_{x \to - 3} \frac{3}{{(x + 3)}^{2}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Infinite Limits

Date Period

Evaluate each limit. Write DNE if the limit does not exist.

$\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$
$\lim_{x \to - 3} \frac{6}{(x + 3)}$
$\lim_{x \to - 1^{-}} \frac{- 2}{(x + 1)}$
$\lim_{x \to - 1^{-}} \frac{- 4}{{(x + 1)}^{2}}$
$\lim_{x \to - 5^{-}} \frac{- 5}{{(x + 5)}^{2}}$
$\lim_{x \to 2} \frac{7}{(x - 2)}$
$\lim_{x \to - 1} \frac{8}{(x + 1)}$
$\lim_{x \to - 4^{+}} \frac{- 4}{(x + 4)}$
$\lim_{x \to 1} \frac{- 3}{(x - 1)}$
$\lim_{x \to 4^{-}} \frac{- 8}{{(x - 4)}^{2}}$
$\lim_{x \to 2} \frac{8}{{(x - 2)}^{2}}$
$\lim_{x \to 4} \frac{1}{(x - 4)}$
$\lim_{x \to - 3} \frac{5}{{(x + 3)}^{2}}$
$\lim_{x \to - 4} \frac{8}{(x + 4)}$
$\lim_{x \to - 4^{+}} \frac{- 1}{(x + 4)}$
$\lim_{x \to 1} \frac{1}{{(x - 1)}^{2}}$
$\lim_{x \to - 3^{-}} \frac{- 3}{{(x + 3)}^{2}}$
$\lim_{x \to 2^{-}} \frac{3}{(x - 2)}$
$\lim_{x \to - 2^{+}} \frac{6}{(x + 2)}$
$\lim_{x \to - 3} \frac{3}{{(x + 3)}^{2}}$



Sigma Prep · sigmaprep.io/worksheets Name Infinite LimitsAnswer keyVersion 1 Date Period Evaluate each limit. Write DNE if the limit does not exist. $\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$ $- \infty$ $\lim_{x \to - 3} \frac{6}{(x + 3)}$ DNE $\lim_{x \to - 1^{-}} \frac{- 2}{(x + 1)}$ $\infty$ $\lim_{x \to - 1^{-}} \frac{- 4}{{(x + 1)}^{2}}$ $- \infty$ $\lim_{x \to - 5^{-}} \frac{- 5}{{(x + 5)}^{2}}$ $- \infty$ $\lim_{x \to 2} \frac{7}{(x - 2)}$ DNE $\lim_{x \to - 1} \frac{8}{(x + 1)}$ DNE $\lim_{x \to - 4^{+}} \frac{- 4}{(x + 4)}$ $- \infty$ $\lim_{x \to 1} \frac{- 3}{(x - 1)}$ DNE $\lim_{x \to 4^{-}} \frac{- 8}{{(x - 4)}^{2}}$ $- \infty$ $\lim_{x \to 2} \frac{8}{{(x - 2)}^{2}}$ $\infty$ $\lim_{x \to 4} \frac{1}{(x - 4)}$ DNE $\lim_{x \to - 3} \frac{5}{{(x + 3)}^{2}}$ $\infty$ $\lim_{x \to - 4} \frac{8}{(x + 4)}$ DNE $\lim_{x \to - 4^{+}} \frac{- 1}{(x + 4)}$ $- \infty$ $\lim_{x \to 1} \frac{1}{{(x - 1)}^{2}}$ $\infty$ $\lim_{x \to - 3^{-}} \frac{- 3}{{(x + 3)}^{2}}$ $- \infty$ $\lim_{x \to 2^{-}} \frac{3}{(x - 2)}$ $- \infty$ $\lim_{x \to - 2^{+}} \frac{6}{(x + 2)}$ $\infty$ $\lim_{x \to - 3} \frac{3}{{(x + 3)}^{2}}$ $\infty$ 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 DNE if the limit does not exist.

$\lim_{x \to 2} \frac{- 5}{{(x - 2)}^{2}}$ $- \infty$
$\lim_{x \to - 3} \frac{6}{(x + 3)}$ DNE
$\lim_{x \to - 1^{-}} \frac{- 2}{(x + 1)}$ $\infty$
$\lim_{x \to - 1^{-}} \frac{- 4}{{(x + 1)}^{2}}$ $- \infty$
$\lim_{x \to - 5^{-}} \frac{- 5}{{(x + 5)}^{2}}$ $- \infty$
$\lim_{x \to 2} \frac{7}{(x - 2)}$ DNE
$\lim_{x \to - 1} \frac{8}{(x + 1)}$ DNE
$\lim_{x \to - 4^{+}} \frac{- 4}{(x + 4)}$ $- \infty$
$\lim_{x \to 1} \frac{- 3}{(x - 1)}$ DNE
$\lim_{x \to 4^{-}} \frac{- 8}{{(x - 4)}^{2}}$ $- \infty$
$\lim_{x \to 2} \frac{8}{{(x - 2)}^{2}}$ $\infty$
$\lim_{x \to 4} \frac{1}{(x - 4)}$ DNE
$\lim_{x \to - 3} \frac{5}{{(x + 3)}^{2}}$ $\infty$
$\lim_{x \to - 4} \frac{8}{(x + 4)}$ DNE
$\lim_{x \to - 4^{+}} \frac{- 1}{(x + 4)}$ $- \infty$
$\lim_{x \to 1} \frac{1}{{(x - 1)}^{2}}$ $\infty$
$\lim_{x \to - 3^{-}} \frac{- 3}{{(x + 3)}^{2}}$ $- \infty$
$\lim_{x \to 2^{-}} \frac{3}{(x - 2)}$ $- \infty$
$\lim_{x \to - 2^{+}} \frac{6}{(x + 2)}$ $\infty$
$\lim_{x \to - 3} \frac{3}{{(x + 3)}^{2}}$ $\infty$

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.

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.

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

Select your difficulty

Infinite Limits

Infinite LimitsAnswer keyVersion 1

How to do these

Where students go wrong

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