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.

Related Topics

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

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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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.