Answer
$h(t)$ does not have a continuous extension at $t=0$. However, $h(t)$ can still be extended to be continuous from either the left or the right at $t=0$.
- To be continuous from the left, $h(t)$ needs to include the value $h(0)=0.368$.
- To be continuous from the right, $h(t)$ needs to include the value $h(0)=2.718$.
Work Step by Step
$$h(t)=(1+|t|)^{1/t}$$
The graph is enclosed below.
Condition to have a continuous extension for $h(t)$ at $t=c$: $\lim_{t\to c}h(t)$ must exist.
But here, $\lim_{t\to0}h(t)$ does not exist, because as $t\to0$ from the left and the right, $h(t)$ approaches different values:
- As $t\to0^+$, $h(t)$ approaches $2.718$.
- As $t\to0^-$, $h(t)$ approaches $0.368$.
So $h(t)$ does not have a continuous extension at $t=0$. However, $h(t)$ can still be extended to be continuous from either the left or the right at $t=0$.
- To be continuous from the left, $h(t)$ needs to include the value $h(0)=0.368$.
- To be continuous from the right, $h(t)$ needs to include the value $h(0)=2.718$.
