l'Hospital Calculus Problem

Compute the following limit

$\lim_{x\rightarrow 0} \frac{\sin{(5x)}}{x}$

Evaluate the following limit

$\lim_{x\rightarrow \infty} \frac{\ln{(1 + e^{3x})}}{2x + 5}$

Explain why the following limit can not be found using l'Hospital's Rule then find the limit using a different method.

$\lim_{x\rightarrow \infty} \frac{x + \cos{(x)}}{x}$

Evalute $\lim_{x\rightarrow 0} \frac{x + \cos{(2x)} - e^x}{x}$