Answer to Question #143598 in Calculus for sachini

Question

if a function f is not defined at x=a that the limit

Answer 1

The concept of limits has to do with the behaviour of the function close to x = a and not at x = a.

$\lim\limits_{x\to a}f(x)$ may exist even if function f is undefined at $x=a$  

$f(x)=x, x\not=0$

$\lim\limits_{x\to 0}f(x)=0,$ $f$ is undefined at $x=0.$

$\lim\limits_{x\to a}f(x)$ may be equal infinity even if function f is undefined at $x=a$  

$f(x)=\dfrac{1}{|x|},$

$\lim\limits_{x\to 0}f(x)=\infin,$ $f$ is undefined at $x=0.$

$\lim\limits_{x\to a}f(x)$ may not exist if function $f$ is undefined at $x=a$  

$f(x)=\dfrac{1}{x},$

$\lim\limits_{x\to 0^-}f(x)=-\infin, \lim\limits_{x\to 0^+}f(x)=\infin,$

$\lim\limits_{x\to 0}f(x)=\text{does not exist }$