Limits Calculator

Published on: April 27, 2025

Enter a function and the point to approach, separated by a comma (e.g., (x^2-1)/(x-1), 1) to compute its limit.

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

Formula:

Formula

lim_x→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 lim_x→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 (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 lim_x→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