## University Calculus: Early Transcendentals (3rd Edition)

$-1$
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$ Here, $\lim\limits_{\theta \to 0} f(0)=\dfrac{0}{0}$ This shows an indeterminate form of the limit, so we need to use L'Hospital's rule. $\lim\limits_{\theta \to 0} \dfrac{-\sin \theta}{e^\theta-1}=\dfrac{0}{0}$ This shows an indeterminate form of the limit, so we need to use L'Hospital's rule. $\lim\limits_{\theta \to 0} \dfrac{-\cos \theta}{e^\theta}=\dfrac{-1}{1}=-1$