Help in evaluating limit of the given function. 
Evaluate the limit $$\lim_{x\to \infty} \left(\cfrac{x^2+5x+3}{x^2+x+2}\right)^x $$

I'm not sure how to evaluate this limit. This is what I've done yet: 
$$\begin{align} 
\lim_{x\to \infty} \left\{1 + \left(\cfrac{x^2+5x+3}{x^2+x+2} - 1\right) \right\} ^x\\ 
=\ \lim_{x\to \infty} \left\{1 + \left(\cfrac{4x+1}{x^2+x+2}\right) \right\}^x\end{align}$$
Not sure where to proceed from here. Out of ideas! 
Any help will be greatly appreciated.

It depends on what machinery you have at your disposal. Can you use the continuity of functions? The rules of de l'Hospital? Logarithms? 
Ultimately, what you have to come to grips with here is that in the limit you have the indeterminate form $1^\infty$, which one normally reduces to one of the simpler forms $0\over0$ or $\infty\over\infty$ before doing the evaluation of the limit.
