Answer
$f$ does not have a continuous extension to the origin. But it still can extend to be continuous at the origin from either the left or the right:
- From the left: the extended function's value is $-1$.
- From the right: the extended function's value is $1$.
Work Step by Step
$$f(x)=\frac{\sin x}{|x|}$$
The graph of the function is shown below.
For $f$ to have a continuous extension to the origin, $\lim_{x\to0}f(x)$ must exist first. Yet, looking at the graph, $f(x)$ approaches a value as $x\to0^+$, and $f(x)$ approaches another value as $x\to0^-$. So $\lim_{x\to0}f(x)$ does not exist, and $f$ does not have a continuous extension to the origin.
But since $\lim_{x\to0^+}f(x)$ and $\lim_{x\to0^-}f(x)$ do exist, $f(x)$ can still be extended to be continuous at the origin from either the left or the right, if $f(0)$ acquires the value of either $\lim_{x\to0^+}f(x)$ or $\lim_{x\to0^-}f(x)$
- To be continuous at the origin from the left: As $x\to0^-$ we see that $f(x)$ approaches $-1$, meaning $\lim_{x\to0^-}f(x)=-1$. So $f(0)$ needs to acquire the value $-1$.
- To be continuous at the origin from the right: As $x\to0^+$ we see that $f(x)$ approaches $1$, meaning $\lim_{x\to0^+}f(x)=1$. So $f(0)$ needs to acquire the value $1$.
