Math Practice

Finding Limits From a Graph Practice Problems

Each problem describes a graph. Trace the curve from each side to find the limit, and look at the filled dot only when the question asks for f(a).

Questions

1.
On a graph of , the curve approaches height 5 from both sides of . There is an open circle at and a filled dot at . Find .
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The limit follows the curve, which heads for the open circle at height 5. The dot doesn’t matter.
2.
For the same graph (open circle at , filled dot at ), find .
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is the height of the filled dot.
3.
The graph of is an unbroken parabola. Find .
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Where the curve has no break, the limit is the height of the curve: .
4.
On a graph, the curve approaches height 4 as from the left and height 1 as from the right. Find .
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The two sides approach different heights, so this is a jump and the two-sided limit does not exist.
5.
A graph has an open circle at and no filled dot anywhere above or below . Both sides of the curve head for the circle. Find , or type undefined.
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With no filled dot at , the function has no value there, even though the limit (0) exists.
6.
For the same graph (open circle at , both sides heading for it), find .
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Both sides approach the circle’s height, 0.
7.
Graph for and for . Find .
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From the left, the graph follows , heading for .
8.
For the same function ( for , for ), find .
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From the right, the graph follows , heading for .
9.
For the same function ( for , for ), find .
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From the left it approaches 3 and from the right 5. They disagree, so the limit does not exist.
10.
Graph for and for , with . Find .
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Both pieces head for height 3 at ( and ). The point is a moved point and doesn’t affect the limit.
11.
A graph has a vertical asymptote at , and the curve rises without bound on both sides of it. Does exist? Give its value, or type DNE.
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No finite height is approached. We may write to describe the behaviour, but the limit does not exist.
12.
The graph of looks like an unbroken wave except for an open circle at . Find .
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Both sides of the curve head for the open circle at height 1, so the limit is 1 even though the function is undefined at 0.

Written by

Zohaib

MPhil in Mathematics, Founder of Mathovia

Published

3 October 2026

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Finding Limits From a Graph Practice Problems

These problems follow the finding limits from a graph lesson. Picture each graph, or sketch it, before you answer.

How to read each graph

Trace from the left towards and note the height.
Trace from the right and note the height.
Compare. The same height is the limit; different heights mean the limit does not exist.
For , look only at the filled dot.

Common mistakes to watch for

Reading the dot instead of the curve. The limit follows where the curve heads; a filled dot only gives .
Checking one side. Trace the curve from both the left and the right.
Calling a jump a limit. If the sides approach different heights, the limit does not exist.
Calling infinity a value. At a vertical asymptote the limit does not exist, even if we write .

Frequently Asked Questions

What do I type when a limit does not exist?+

Type DNE. Use it for jumps, where the two sides approach different heights, and for vertical asymptotes, where the curve grows without bound.

Why do some answers say undefined?+

Those questions ask for the value f(a), not the limit. If there is no filled dot at x = a, the function has no value there.

limFinding Limits From a Graph Practice Problems

Finding Limits From a Graph Practice Problems

How to read each graph

Common mistakes to watch for

Frequently Asked Questions

More practice problems

Finding Limits From a Graph Practice Problems