Answer
$k(x)$ does not have a continuous extension at $x=0$. However, $k(x)$ can still be extended to be continuous from either the left or the right at $x=0$.
- To be continuous from the left, $k(x)$ needs to include the value $k(0)=-1.443$.
- To be continuous from the right, $k(x)$ needs to include the value $k(0)=1.443$.
Work Step by Step
$$k(x)=\frac{x}{1-2^{|x|}}$$
The graph is enclosed below.
Condition to have a continuous extension for $k(x)$ at $x=c$: $\lim_{x\to c}k(x)$ must exist.
But here, $\lim_{x\to0}k(x)$ does not exist, because as $x\to0$ from the left and the right, $k(x)$ approaches different values:
- As $x\to0^+$, $k(x)$ approaches $1.443$.
- As $x\to0^-$, $k(x)$ approaches $-1.443$.
So $k(x)$ does not have a continuous extension at $x=0$. However, $k(x)$ can still be extended to be continuous from either the left or the right at $x=0$.
- To be continuous from the left, $k(x)$ needs to include the value $k(0)=-1.443$.
- To be continuous from the right, $k(x)$ needs to include the value $k(0)=1.443$.