Limit to infinity question 
Possible Duplicate:
Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials 

Does $\lim_{x\rightarrow \infty} \frac{5x}{(1+x^2)} = 0$ or $\lim_{x\rightarrow \infty} \frac{5x}{(1+x^2)} = 1$?
I am asking because I was wondering if $\infty^2$ at the denominator is "considered" bigger than $5\infty$. Or do we just take the above as $\frac{\infty}{\infty}$?

You have (dividing through by $x$)
$$\lim_{x\to \infty} \frac{5x}{1+x^2} = \lim_{x\to \infty} \frac{5}{x^{-1} + x} = 0.$$
Of if you know about L'Hopital's rule:
$$\lim_{x\to \infty} \frac{5x}{1+x^2} = \lim_{x\to \infty} \frac{5}{2x} = 0.$$
Note that in general we can't talk about $\frac{\infty}{\infty}$. $\infty$ is not a number so we can't really use it as such. Usually in calculus when we talk about infinity we think of it in terms of limits.
