Limits – Sam Artigliere's Blog

The limit $\displaystyle\lim_{x\to c} f(x) = L$ exists if

if, for any $\epsilon>0$ and $0< \lvert f(x)-L \rvert < \epsilon$
there exist $\delta>o$ such that $0 < \lvert x-c \rvert < \delta$

Stated in words, no matter how close $f(x)$ gets to some value $f(c)=L$ , you can always find a value of $x$ that’s very close to $x=c$ .

Said another way, as $x$ gets closer and closer to some number $c$ (without actually getting to $c$ ), $f(x)$ gets closer and closer to some value (without actually reaching that value). That value, $f(c)=L$ that $f(x)$ is approaching is called the limit of $f(x)$ as $x$ approaches $c$ .

Notice the phrases “without actually getting to” and “without actually reaching”. This implies that a function can have a limit without that limit actually being a value of a function. To illustrate this, in the diagram, $f(x)$ is purposely shown to be discontinuous at $f(c)=L$ , yet $\displaystyle\lim_{x\to c} f(x)$ still exists (i.e., its value is $L$ ).