Limits and Continuity Explained

Limits are the foundation upon which all of calculus is built. Before you can understand derivatives or integrals, you need to understand what a limit is and how to evaluate one.

What Is a Limit?

The limit of f(x) as x approaches a is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a. It does not matter what f(a) actually is, or even if f(a) is defined. The limit only cares about behavior near a, not at a.

Evaluating Limits

For most functions, you can just plug in the value. But sometimes this gives an indeterminate form like 0/0. In those cases, you factor, simplify, multiply by the conjugate, or use L’Hôpital’s rule (if you know derivatives).

Important limits to memorize: lim(x→0) sin(x)/x = 1, lim(x→∞) (1+1/x)^x = e, lim(x→0) (e^x-1)/x = 1.

Continuity

A function is continuous at a point if three conditions hold: f(a) is defined, the limit as x approaches a exists, and the limit equals f(a). A function is continuous on an interval if it is continuous at every point in that interval.

Use our Limit Calculator to evaluate common limits.