Hi I need to prove
lim (x-3)/(x^{2}+1) as x goes to infinity is zero using the formal definitional. Can anyone help?
Why in the world would you even use L'Hopital? If you could use any method, it's obvious that the denominator is going to infinity faster than the numerator.
However, since you are required to use the "formal definition", You want to show that, for any $\displaystyle \epsilon> 0$, we can find $\displaystyle \delta> 0$ such that if $\displaystyle x> \delta$, then $\displaystyle \left|\frac{x- 3}{x^2+ 1}\right|< \epsilon$.
We can start by writing that as $\displaystyle -\epsilon< \frac{x- 3}{x^2+ 1}\right|< \epsilon$. It is certainly true that $\displaystyle x^2+ 1> 0$ for all x so that is the same as $\displaystyle -\epsilon(x^2+ 1)< x- 3< \epsilon(x^2+ 1)$. And then $\displaystyle \epsilon x^2- x+ \epsilon+ 3> 0$. Solve that quadratic equation and keep the larger root to see how large x must be in order that all this (which is reversible so you can go back to $\displaystyle \left|\frac{x- 3}{x^2+ 1}\right|< \epsilon$) be true.