Answer
$f(x)$ can chooses to extend to be either continuous from the left or from the right at the origin:
- From the left: the extended function's value should be $-2.3$.
- From the right: the extended function's value should be $2.3$.
Work Step by Step
$$f(x)=\frac{10^{|x|}-1}{x}$$
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: Zooming in the graph, we see that $\lim_{x\to0^-}f(x)=-2.3$. So $f(0)$ needs to acquire the value $-2.3$.
- To be continuous at the origin from the right: Zooming in the graph, we see that $\lim_{x\to0^+}f(x)=2.3$. So $f(0)$ needs to acquire the value $2.3$.
