# Limit at Infinity

#### DanSlevin

##### New member
Hello, I am working on some limit problems and ran into one in which I am lost on how to proceed with:

The problem is:

lim
x -> infinity

x^(2/3)
x/(log^2(x))

I have a basic understanding of L'Hopital's Rule and attempted to apply it, but just ended up with a confusing mess. I'm assuming I'm supposed to simply first, but am not sure how to proceed. Any guidance as to the techniques to use here would be great. Thanks.

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#### Prove It

##### Well-known member
MHB Math Helper
Hello, I am working on some limit problems and ran into one in which I am lost on how to proceed with:

The problem is:

lim
x -> infinity

x^(2/3)
x/(log^2(x))

I have a basic understanding of L'Hopital's Rule and attempted to apply it, but just ended up with a confusing mess. I'm assuming I'm supposed to simply first, but am not sure how to proceed. Any guidance as to the techniques to use here would be great. Thanks.
Is it this?

$\displaystyle \lim_{x \to \infty}\frac{x^{\frac{2}{3}}}{\frac{x}{(\log{x})^2}}$

#### DanSlevin

##### New member
I'm sorry about that. I'm also having some trouble understanding the Latex syntax so hopefully this makes it a little clearer.

x2/3
________
x/log2(x)

as x approaches infinity.

#### Also sprach Zarathustra

##### Member
I'm sorry about that. I'm also having some trouble understanding the Latex syntax so hopefully this makes it a little clearer.

x2/3
________
x/log2(x)

as x approaches infinity.

$$\Large \lim_{x \to \infty}\frac{x^{\frac{2}{3}}}{\frac{x}{(\log{x})^2 }}$$

$$\Large \lim_{x \to \infty}\frac{(\log{x})^2 }{x^{\frac{1}{3}}}$$

Now, try to use l'Hôpital's rule.

Last edited by a moderator:

#### DanSlevin

##### New member
Thank you for your reply and answer. Do you happen to have any sites I could look at that include examples of simplifications of this manner? I'm unfortunately unable to find any that show examples with fractional exponents and I'm a little confused as to some of the steps. Thanks again.

#### CaptainBlack

##### Well-known member
Thank you for your reply and answer. Do you happen to have any sites I could look at that include examples of simplifications of this manner? I'm unfortunately unable to find any that show examples with fractional exponents and I'm a little confused as to some of the steps. Thanks again.
The usual laws of exponents apply for all exponents be they integer, common fractions or decimals.

CB