Answer
$1$
Work Step by Step
Here, we have $\lim\limits_{a \to \infty}\int_0^{a} e^{-\theta} \sin \theta d\theta=\lim\limits_{a \to \infty}[ -e^{-\theta} (\cos \theta+\sin \theta)|_0^{a}$
This implies that
$\lim\limits_{a \to \infty}[ -e^{-\theta} (\cos \theta+\sin \theta)|_0^{a}=\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a-(-e^{-0} (\cos 0+\sin 0)]$
or, $\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a-(-e^{-0} (\cos 0+\sin 0)]=\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a)+1]$
Thus,
$\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a)+1]=0+1=1$
