Calculators / Calculus 1 / Limits Calculator

Limits Calculator

Published on: April 27, 2025

This Limits Calculator helps you evaluate the limit of a function as the input approaches a given value and shows each step clearly. It can be used to work through common limit problems by simplifying expressions, substituting values where possible, and applying standard limit rules. This makes it useful for checking answers, understanding how limits work, and practising calculus step by step.

Step-by-step method

  1. Write the limit in standard form.
  2. Substitute x = a.
  3. If substitution is indeterminate (like 0/0), simplify the function and substitute again.
  4. State the final limit.

Formula:

Formula

limx→af(x)

Example 1: f(x) = (x^2 - 1)/(x - 1), a = 1

Step 1 - Write the limit.

In this problem: Use standard limit notation.

Compute limx→1
x2 − 1
x − 1

Step 2 - Substitute x = a.

In this problem: If you get 0/0 (indeterminate), simplify/cancel factors.

f( 1 )=
( 1 )2 − 1
( 1 ) − 1
=
0
0
=0/0 (indeterminate)

Step 3 - Simplify the function.

In this problem: Factor and cancel common factors.

x2 − 1
x − 1
=
(x + 1)(x − 1)
x − 1
=x + 1

Step 4 - Substitute again.

In this problem: Now substitution is defined.

f( 1 )=( 1 ) + 1=2

Step 5 - State the limit.

In this problem: After simplification, substitution worked.

Limit=2

Final answer: Limit = 2

Example 2: f(x) = x^2 + 3x, a = 2

Step 1 - Write the limit.

In this problem: Use standard limit notation.

Compute limx→2 x2 + 3x

Step 2 - Substitute x = a.

In this problem: Direct substitution is defined.

f( 2 )=( 2 )2 + 3( 2 )=10

Step 3 - State the limit.

In this problem: Direct substitution worked.

Limit=10

Final answer: Limit = 10