Answer
$f(x)$ has a continuous extension to the origin. The extended function's value at $x=0$ is $2.3$
Work Step by Step
$$f(x)=\frac{10^x-1}{x}$$
The graph of the function is shown below.
Looking at the graph, $f(x)$ is not defined at $x=0$, but there is a good chance that we can extend $f(x)$ to include a value of $x=0$ so that $f(x)$ is continuous at $x=0$.
The reason is that as $x$ approaches $0$ from the both the left and the right, $f(x)$ approaches a value of $2.3$. In other words, $\lim_{x\to0}f(x)=2.3$. And for $f(x)$ to be continuous at $x=0$, $f(0)$ just needs to acquire a value of $2.3$ as well.
Therefore, $f(x)$ has a continuous extension to the origin. The extended function's value at $x=0$ is $2.3$.
