Proving limits using epsilon definition I want to prove that $$\lim_{x\to\infty}\frac{x}{x^{2}+1} = 0.$$ So I start by saying, given $\varepsilon>0$ I want to find $M>0$ such that $$\forall x>M\implies\left|\frac{x}{x^{2}+1}-0\right|<\varepsilon.$$ Now everyone who has answered me so far has plucked an $M$ out of thin air without any indication as to how they have found it. So assuming we don't know $M$ how would we find it?

The trick is to make the expression easier to deal with. For example, for $x > 0$

$$\left| \frac{x}{x^2 + 1} - 0 \right| \leq \left| \frac{x}{x^2} \right| = \frac{1}{x} $$

Now it becomes straight forward to find an appropriate $M$ given $\epsilon > 0$.