Answer
$0$
Work Step by Step
Consider: $\lim\limits_{t \to 0}f(t)=\lim\limits_{t \to 0}\dfrac{\sin t^2}{t}$
Need to check that the limit has an indeterminate form.
Thus, $f(0)=\dfrac{\sin 0}{0}=\dfrac{0}{0}$
Now, apply L-Hospital's rule such as: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$
This implies:
$\lim\limits_{t \to 0}\dfrac{(2t) \cos t^2}{1}= \lim\limits_{t \to 0}\dfrac{ 2(0) \cos 0}{1}=0$
