Limit

For a real function $f(x)$ and some value $c$ , $\lim_{x\rightarrow c} f(x)$ (said, "the limit of $f$ at $x$ as $x$ goes to $c$ ) equals $L$ iff for every $\epsilon > 0$ there exists a $\delta$ such that if $0<|x-c|<\delta$ , then $|f(x)-L|< \epsilon$ .

Intuitive Meaning

The definition of a limit is a difficult thing to grasp, so many books give an intuitive definition first: a limit is the value to which the rest of the function grows closer. For example, $\lim_{x\rightarrow 2}x^2=4$ , because as $x$ grows arbitrarily close to 2 from either direction, the function $f(x)=x^2$ grows arbitrarily close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is, $\lim_{x\rightarrow c} f(x) = f(c)$ . This is because the function we chose was a continuous function. Unfortunately, this does not hold true in general. For example, consider the function $f(x)$ over the reals defined to be 0 if $x\neq 0$ and 1 if $x=0$ . Although the value of the function at 0 is 1, the limit $\lim_{x\rightarrow 0}f(x)$ is, in fact, zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, $f(x)$ will always be close to (in fact equal to) zero. Note that if our definition required only that $|x-c|<\delta$ , the limit of this function would not exist.

Existence of Limits

Limits do not always exist. For example $\lim_{x\rightarrow 0}\frac{1}{x}$ does not exist, since, in fact, there exists no $\epsilon$ for which there exists $\delta$ satisfying the definition's conditions, since $\left|\frac{1}{x}\right|$ grows arbitrarily large as $x$ approaches 0. However, it is possible for $\lim_{x\rightarrow c} f(x)$ not to exist even when $f$ is defined at $c$ . For example, consider the Dirichlet function, $D(x)$ , defined to be 0 when $x$ is irrational, and 1 when $x$ is rational. Here, $\lim_{x\rightarrow c}D(x)$ does not exist for any value of $c$ . Alternatively, limits can exist where a function is not defined, as for the function $f(x)$ defined to be 1, but only for nonzero reals. Here, $\lim_{x\rightarrow 0}f(x)=1$ , since for $x$ arbitrarily close to 0, $f(x)=1$ .