Answer
See explanation
Work Step by Step
Let's describe several ways in which a limit can fail to exist. Illustrate with sketches.
The given definition of a limit is: $\lim\limits_{x \to a}f(x)=L$ if and only if $\lim\limits_{x \to a^-}f(x)= \lim\limits_{x \to a^+}f(x)=L$. Thus the limit will fail if $\lim\limits_{x \to a^-}f(x)\neq\lim\limits_{x \to a^+}f(x)$. There are several ways for this to occur but here are some examples:
1) The left-hand and right-hand limits are not equal (jump discontinuity)
$\lim\limits_{x \to a^-}f(x)= L_1$ and $\lim\limits_{x \to a^+}f(x)=L_2$ where $L_1 \neq L_2$
2) The function increases or decreases without bound (infinite discontinuity)
$\lim\limits_{x \to a^-}f(x)= \infty$ and $\lim\limits_{x \to a^+}f(x)=-\infty$ creating an infinite discontinuity.
3) The function oscillates endlessly (oscillatory discontinuity)
The function keeps swinging between values and never settles.
Example: $f(x)=\sin \frac{1}{x}$ as $x\rightarrow 0$
