# Delta Epsilon Proof Help

• Feb 19th 2013, 05:09 PM
skatejawn
Delta Epsilon Proof Help
Hi I need to prove

lim (x-3)/(x2+1) as x goes to infinity is zero using the formal definitional. Can anyone help?
• Feb 19th 2013, 05:16 PM
peruvian
Re: Delta Epsilon Proof Help
Sucks I would L'Hop the out of it
• Feb 19th 2013, 06:35 PM
HallsofIvy
Re: Delta Epsilon Proof 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.