How to evaluate this limit/simplify the expression?

Question

I need help with the limit $$\lim_{x \to \infty} \frac{1+\cos^2 x-\sin^2 x}{e^{\sin x}\cos x(x+\cos x\sin x) + e^{\sin x}(1+\cos^2 x-\sin^2 x)}$$ (It evaluates to zero.) How do I simplify this expression? Thank you for your help.

Answer 1

When $\cos x=0,$ that is, when $x$ is an odd multiple of $\pi/2,$ the expression is of the form $\frac{0}{0},$ so we have to assume it is defined by continuity at these points. This will work out to be $0$. With that understanding, $$ \lim_{x \to \infty} \frac{1+\cos^2 x-\sin^2 x}{e^{\sin x}\cos x(x+\cos x\sin x) + e^{\sin x}(1+\cos^2 x-\sin^2 x)}=\\ \lim_{x \to \infty}\frac{2\cos^2 x}{e^{\sin x}\cos x(x+\cos x\sin x) + 2e^{\sin x}\cos^2 x}=\\ \lim_{x \to \infty}\frac{2\cos x}{e^{\sin x}(x+\cos x\sin x+2\cos x)}=0, $$ since the numerator is bounded and the denominator goes to $\infty$ as $x\to\infty.$