Let $f$ be a function defined everywhere in an open interval containing $c$, except possibly at $c$. Then the limit of the function $f$ as $x$ approaches $c$ is the number $L$, written $ \lim_{x \rightarrow c} f(x)=L $ if, given any number $\epsilon>0$, there is a number $\delta>0$ so that $ \text { whenever } 0<|x-c|<\delta \text { then }|f(x)-L|<\epsilon $ --- In other words, $\delta$ is how far left and right one can look from $x=c$ to find points that do not go more thn $\epsilon$ above or below the value of the limit $L$. The idea is that as $\epsilon$ gets smaller, $\delta$ will tend to get smaller too. If that doesn't happen, then the limit doesn't exist --- from (Sullivan, Miranda) - The limit $L$ of a function $y=f(x)$ as $x$ approaches a number $c$ does not depend on the value of $f$ at $c$. - The limit $L$ of a function $y=f(x)$ as $x$ approaches a number $c$ is unique; that is, a function cannot have more than one limit. (A proof of this property is given in Appendix B.) - If there is no single number that the value of $f$ approaches as $x$ gets close to $c$, we say that $f$ has no limit as $x$ approaches $c$, or more simply, that the limit of $f$ does not exist at $c$.