Functions
AI Assistant

What is a Limit?

Approaching a value — from the left, from the right, and from both sides

What value does f(x) = \frac{x^2 - 1}{x - 1} approach as x gets close to 1? You can't plug in x = 1 (you'd get 0/0). But approaching from both sides, the function gets closer to 2. That's the limit.

A limit asks: "What does f(x) GET CLOSE TO as x approaches some value?" The function doesn't need to REACH that value — it might have a hole there.

In this lesson you'll see a function with a hole, watch a point approach it, and discover what limits really mean.

Graph

FAQ

What is a limit?
A limit is the value a function approaches as the input approaches a specific value. Written \lim_{x \to a} f(x) = L.
Why can't I just plug in the value?
Sometimes plugging in gives 0/0 — an indeterminate form. The function may still approach a definite value. Factor and simplify first.
What is a hole in a graph?
A hole (removable discontinuity) occurs when a function is undefined at a point but has a limit there. The graph looks continuous except for a single missing dot.
What are one-sided limits?
One-sided limits approach from only one direction. The two-sided limit exists only if both one-sided limits are equal.

Related Topics

What is a Derivative?What is a Function?Quadratic Equations