Limit

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

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, $\displaystyle\lim_{x\rightarrow 2}x=2$ , because as the function $x$ grows arbitrarily close to 2 from either direction, the function $\displaystyle f(x)=x$ grows arbitrarily close to 2. However, although in this case, and in many others, the limit of $\displaystyle f(x)$ as $\displaystyle x$ goes to $\displaystyle c$ equals $\displaystyle f(c)$ , this is not always true. For example, consider the function $\displaystyle f(x)$ over the reals defined to be 0 if $\displaystyle x\neq 0$ , and 1 if $\displaystyle x=0$ . Although it might seem as though $\displaystyle \lim_{x\rightarrow 0}f(x)$ would equal 1, it is in fact zero, because no matter how close we get to zero, as long as we never actually reach zero, $\displaystyle f(x)$ will always be less than zero. This is the function of the strict inequality in $\displaystyle 0<|x-\delta|$ .