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: Limits describe behavior near a point, even if the function isn't defined there.

Common stuck point: The limit can exist even if f(a) doesn't. It's about approaching, not reaching.

Sense of Study hint: Try plugging in values very close to the target from both sides and see what output they approach.

Worked Examples

Example 1

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

Solution

  1. 1
    Since 2x + 1 is a polynomial, it is continuous everywhere.
  2. 2
    For continuous functions, we can evaluate the limit by direct substitution.
  3. 3
    Substitute x = 3: 2(3) + 1 = 6 + 1 = 7.

Answer

\lim_{x \to 3} (2x + 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 \to 2} \frac{x^2 - 4}{x - 2}

Example 3

hard
Find \lim_{x \to 0} \frac{\sin x}{x}

Practice Problems

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

Example 1

easy
Find \lim_{x \to 5} (x^2 - 3x + 2)

Example 2

medium
Find \lim_{x \to -1} \frac{x^2 + 3x + 2}{x + 1}
← Back to Limit Formula Explained Practice Mode

Background Knowledge

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

function definition