Find the limit of a function as x approaches a value, with the one-sided limits shown separately.
The method behind this calculator
The calculator samples the function at points progressively closer to your target from each side. That reveals removable discontinuities — where the limit exists but the function is undefined at the point itself — which is exactly the case most limit problems are built around.
When the left and right limits disagree, the two-sided limit does not exist, and the calculator says so rather than averaging them into a misleading number.
How to use it
- Enter a function of x.
- Enter the value x approaches.
- Choose two-sided, left or right.
The calculation
the limit exists when the left and right limits agree
f is continuous at a when lim(x→a) f(x) = f(a)
The default scenario, worked through
The calculator opens with a function f of (x^2 – 1)/(x – 1). On those figures the limit is 2. Underneath, Left-hand limit comes out at 2 and Right-hand limit at 2. Change any field and every figure updates as you type.
Frequently asked questions
Why does 0/0 have a limit?
Because the limit describes behaviour near the point, not at it. (x²−1)/(x−1) is undefined at x = 1 but approaches 2 from both sides.
What is a one-sided limit?
The value approached from only one direction. Step functions and square roots at their domain edge often have one-sided limits but no two-sided one.
Is this method exact?
It is numerical, so results are precise to several digits rather than symbolically exact. Very oscillatory functions can defeat it.