Answer
$3$
Work Step by Step
Consider: $\lim\limits_{t \to 0}f(t)=\lim\limits_{t \to 0}\dfrac{\ln (t(1-\cos t)}{t-\sin t}$
We need to check that the limit has an indeterminate form.
Thus, $f(0)=\dfrac{0}{0}$
The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$
Then
$\lim\limits_{t \to 0}\dfrac{1-\cos t+t \sin t}{1-\cos t}=\dfrac{0}{0}$
Need to apply L-Hospital's rule again.
$\lim\limits_{t \to 0}\dfrac{3\cos t-t \sin t}{\cos t}=\dfrac{3\cos 0-0}{\cos 0}=3$
