Practice Limit in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.

What output do you get closer and closer to as you get closer and closer to some input?

Showing a random 20 of 50 problems.

Example 1

medium
Evaluate lim⁡x→1x3−1x−1\lim_{x \to 1} \frac{x^3 - 1}{x - 1}.

Example 2

easy
Evaluate lim⁡x→57\lim_{x \to 5} 7.

Example 3

medium
Evaluate lim⁡h→0(2+h)2−4h\lim_{h \to 0} \frac{(2 + h)^2 - 4}{h}.

Example 4

medium
Evaluate lim⁡x→∞3x2+52x2−x+1\lim_{x \to \infty} \frac{3x^2 + 5}{2x^2 - x + 1}.

Example 5

easy
Find lim⁡x→2+(x−2)\lim_{x \to 2^+} (x - 2) and state whether it matches the left-hand limit.

Example 6

medium
Evaluate lim⁡x→4x−4x−2\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}.

Example 7

medium
Evaluate lim⁡x→0x2+2xx\lim_{x \to 0} \frac{x^2 + 2x}{x}.

Example 8

easy
Evaluate lim⁡x→4x\lim_{x \to 4} \sqrt{x}.

Example 9

medium
Find lim⁡x→2x2−4x−2\lim_{x \to 2} \frac{x^2 - 4}{x - 2}

Example 10

easy
Evaluate lim⁡x→0xx+1\lim_{x \to 0} \frac{x}{x + 1}.

Example 11

challenge
Evaluate lim⁡x→∞(x2+x−x)\lim_{x \to \infty} \left( \sqrt{x^2 + x} - x \right).

Example 12

medium
Evaluate lim⁡x→0tan⁡xx\lim_{x \to 0} \frac{\tan x}{x}.

Example 13

medium
Evaluate lim⁡x→2x2−5x+6x−2\lim_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}.

Example 14

hard
Evaluate lim⁡x→∞(x2+x−x)\lim_{x \to \infty} \left(\sqrt{x^2 + x} - x\right).

Example 15

medium
For f(x)={x+1x<23x−3x≥2f(x) = \begin{cases} x+1 & x < 2 \\ 3x-3 & x \ge 2 \end{cases}, find lim⁡x→2f(x)\lim_{x \to 2} f(x).

Example 16

hard
Evaluate lim⁡x→0ex−1x\lim_{x \to 0} \frac{e^x - 1}{x}.

Example 17

medium
Evaluate lim⁡x→∞5x+3x2+1\lim_{x \to \infty} \frac{5x + 3}{x^2 + 1}.

Example 18

medium
Find lim⁡x→−1x2+3x+2x+1\lim_{x \to -1} \frac{x^2 + 3x + 2}{x + 1}

Example 19

medium
Evaluate lim⁡x→0sin⁡5xx\lim_{x \to 0} \frac{\sin 5x}{x}.

Example 20

challenge
Evaluate lim⁡x→0tan⁡x−sin⁡xx3\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}.
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