Limit Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Limit.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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?

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A limit names the single output a function approaches as the input closes in on a target, whether or not the function is defined there.

Common stuck point: The procedure for limit is the easy part; the trap is just substituting aa and stopping when you get 00\frac{0}{0}. Asking "Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?

Worked Examples

Example 1

easy
Find lim⁡x→3(2x+1)\lim_{x \to 3} (2x + 1)

Answer

lim⁡x→3(2x+1)=7\lim_{x \to 3} (2x + 1) = 7

First step

1
Since 2x+12x + 1 is a polynomial, it is continuous everywhere.

Full solution

  1. 2
    For continuous functions, we can evaluate the limit by direct substitution.
  2. 3
    Substitute x=3x = 3: 2(3)+1=6+1=72(3) + 1 = 6 + 1 = 7.
When a function is continuous at a point, the limit equals the function value at that point. Polynomials are continuous everywhere, so direct substitution always works for polynomial limits.

Example 2

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

Example 3

hard
Find lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

hard
Evaluate lim⁡x→0ln⁡(1+x)x\lim_{x \to 0} \frac{\ln(1 + x)}{x}.

Example 9

hard
Use the squeeze theorem to evaluate lim⁡x→0x2sin⁡(1/x)\lim_{x \to 0} x^2 \sin(1/x).

Example 10

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find lim⁡x→5(x2−3x+2)\lim_{x \to 5} (x^2 - 3x + 2)

Example 2

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

Example 3

easy
Evaluate lim⁡x→3(2x+1)\lim_{x \to 3} (2x + 1).

Example 4

easy
Evaluate lim⁡x→2x2\lim_{x \to 2} x^2.

Example 5

easy
Evaluate lim⁡x→0(5−x)\lim_{x \to 0} (5 - x).

Example 6

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

Example 7

easy
Evaluate lim⁡x→1x+3x+1\lim_{x \to 1} \frac{x+3}{x+1}.

Example 8

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

Example 9

easy
Evaluate lim⁡x→0cos⁡x\lim_{x \to 0} \cos x.

Example 10

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

Example 11

medium
Evaluate lim⁡x→3x2−9x−3\lim_{x \to 3} \frac{x^2 - 9}{x - 3}.

Example 12

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

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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 19

challenge
Evaluate lim⁡x→01+x−1x\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}.

Example 20

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

Example 21

challenge
Evaluate lim⁡x→01−cos⁡xx2\lim_{x \to 0} \frac{1 - \cos x}{x^2}.

Example 22

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

Example 23

easy
Evaluate lim⁡x→0(x2+3x+4)\lim_{x \to 0} (x^2 + 3x + 4).

Example 24

easy
Evaluate lim⁡x→3x2−9x−3\lim_{x \to 3} \frac{x^2 - 9}{x - 3}.

Example 25

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

Example 26

easy
Evaluate lim⁡x→0ex\lim_{x \to 0} e^x.

Example 27

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

Example 28

medium
Evaluate lim⁡x→01−cos⁡xx2\lim_{x \to 0} \frac{1 - \cos x}{x^2}.

Example 29

medium
Evaluate lim⁡x→0+1x\lim_{x \to 0^+} \frac{1}{x}.

Example 30

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

Example 31

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

Example 32

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

Example 33

hard
Evaluate lim⁡x→0(1+2x)1/x\lim_{x \to 0} \left(1 + 2x\right)^{1/x}.

Example 34

hard
Evaluate lim⁡x→∞(1+3x)x\lim_{x \to \infty} \left(1 + \tfrac{3}{x}\right)^x.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition