Overview
A limit describes the value a function approaches as its input approaches a point, without necessarily reaching it. The limit may exist even when the function is undefined there, as with sin(x)/x at zero. The calculator checks both one-sided limits because a two-sided limit requires them to agree.
Formula
lim(x→a) f(x) = L, and lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = L.
How it works
Write the function
Enter the mathematical definition in terms of x.
Set the approach point
Enter a number, inf or -inf.
Choose the direction
From both sides or from one side.
Read the result
The one-sided limits, status and continuity check appear.
Worked examples
Example 1: lim(x→0) sin(x)/x
The limit is 1. The function is undefined at zero, so there is a removable discontinuity.
Example 2: lim(x→1) (x² − 1)/(x − 1)
After simplification the expression is x + 1, so the limit is 2.
Example 3: lim(x→+∞) 1/x
The limit is 0 because the fraction approaches zero as x grows.
Frequently asked questions
What is the difference between a limit and a function value?
A limit describes the value being approached, while the function value is the output at the point itself, if defined.
How do I know a limit does not exist?
The one-sided limits differ, or the values fail to approach a finite number.
Does the calculator support infinity?
Yes, enter inf for positive infinity or -inf for negative infinity.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.