Epsilon Delta proof containing decimal exponents

Question

so I have to prove

lim x->infinity ((x^0.8)/(1+x^0.9)) = 0

I am just introduced to epsilon delta, and have no idea how to do this. Please help :( Thanks!

Answer 1

A lot of times when trying to prove a limit exists there are distracting terms in the expression which you can safely ignore. In this case it's the constant one in the denominator.

Explicitly,

$\dfrac{x^{.8}}{1+x^{.9}} \lt \dfrac{x^{.8}}{x^{.9}} =\dfrac1{x^{.1}} $

And now it is easy to choose an $x$ to make this less than $\epsilon$.

The fact that this is not the best possible value does not matter - you just need to show that the expression gets as small as you want.

This is an example of one of my favorite sayings:

Good enough is good enough.