Limit of Fraction $$\lim_{x \to \infty} \frac{(1 + x)^{x/(1 + x)}\cos^{4}x}{e^{x}}$$
Attempt: I've tried evaluating the limits of the terms individually using the property of limits. Also, $y=1/x$ subsituition hasn't helped me, any help will be appreciated. 

Note that  $(1+x)^{\frac{x}{1+x}}=(1+x)(1+x)^{\frac{-1}{1+x}}$ and since $(1+x)^{\frac{-1}{1+x}}\to 1$ as $x\to \infty $ there is an $M>0$ such that
$0\leq \left | \frac{\left ( 1+x \right )^{\frac{x}{1+x}}cos^{4}x}{e^{x}} \right |<2(1+x)e^{-x}$ whenever $x>M$ so that as $x\to \infty $ the limit is $0$
