Answer
$1$
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
Here, $\lim\limits_{x \to 0} f(0)=\lim\limits_{x \to 0}\dfrac{1-\cos x-\cos x \sin x}{\sin 0}=\dfrac{0}{0}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
$\lim\limits_{x \to 0} \dfrac{\sin x+\cos 2x}{\cos x}=\dfrac{0+1}{1}=1$