A limit describes what a function approaches when its input approaches a particular value. The value of the function at the point itself may be different or may not exist at all.

We write

$$\lim_{x\to a}f(x)=L$$

when values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

A value can be missing

Consider

$$f(x)=\frac{x^2-1}{x-1}.$$

For $x\ne1$ this simplifies to $x+1$, so as $x$ approaches $1$ the function approaches $2$:

$$\lim_{x\to1}\frac{x^2-1}{x-1}=2.$$

The original expression is undefined at $x=1$, but the limit still exists. Limits therefore describe nearby behavior rather than simply evaluating the function.

One-sided limits

Sometimes the behavior depends on the direction of approach. The notation

$$\lim_{x\to a^-}f(x)$$

means approaching $a$ from values smaller than $a$, while

$$\lim_{x\to a^+}f(x)$$

means approaching from larger values. A two-sided limit exists only when the two one-sided limits agree.

Limits at infinity

Limits can also describe long-term behavior. For example,

$$\lim_{x\to\infty}\frac1x=0.$$

Here the input does not approach a finite point; instead, the function approaches $0$ as $x$ grows without bound.

Limit laws

When the relevant limits exist, limits respect ordinary arithmetic. For example,

$$\lim(f+g)=\lim f+\lim g,$$

and products and quotients can be handled similarly, provided the limiting denominator is nonzero. These laws reduce many limit problems to simpler ones.