Answer
Yes, it need not be zero.
Work Step by Step
Step 1. Based on the given conditions, we know $g(t)$ and $h(t)$ are defined and $g(0)=h(0)=0$
Step 2. For a new function defined as $k(t)=g(t)/h(t)$, try to find the limit:
$\lim_{t\to0}k(t)=\lim_{t\to0}\frac{g(t)}{h(t)}=\frac{g(0)}{h(0)}=\frac{0}{0}$
Step 3. We have a situation that it is possible for $\lim_{t\to0}k(t)$ to exist, but not necessarily to be zero. For example, let $g(t)=sin(t)$ and $h(t)=t$, we have $g(0)=h(0)=0$ and $k(t)=\frac{sin(t)}{t}$ which leads to $\lim_{t\to0}k(t)=\lim_{t\to0}\frac{sin(t)}{t}=1$
