Solving Limit at Infinity Problem: x^(2/3) / (log^2(x))

In summary, the problem is:limx -> infinityx^(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.
  • #1
DanSlevin
7
0
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.
 
Last edited:
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  • #2
Hard to read, I don't get very well what you mean.

This could be useful: http://www.mathhelpboards.com/forumdisplay.php?26-LaTeX-Help
 
  • #3
DanSlevin said:
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?

\[ \displaystyle \lim_{x \to \infty}\frac{x^{\frac{2}{3}}}{\frac{x}{(\log{x})^2}} \]
 
  • #4
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.
 
  • #5
DanSlevin said:
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:
  • #6
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.
 
  • #7
DanSlevin said:
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
 

FAQ: Solving Limit at Infinity Problem: x^(2/3) / (log^2(x))

1. What is the definition of a limit at infinity?

The limit at infinity refers to the value that a function approaches as its input variable (usually denoted as x) becomes very large, or approaches infinity. This can be thought of as the behavior of the function at the "end" of its domain.

2. How do I solve a limit at infinity problem?

To solve a limit at infinity problem, you can use the following steps: 1) Determine the degree of the numerator and denominator of the function, 2) Divide both the numerator and denominator by the highest power of x, 3) Evaluate the limit using the resulting simplified function. If the resulting function is of the form ∞/∞ or 0/0, you can use algebraic manipulation or L'Hôpital's rule to further simplify the function and evaluate the limit.

3. What is the degree of x^(2/3) / (log^2(x))?

The degree of a function is determined by the highest power of the variable in the function. In this case, the degree of x^(2/3) is 2/3 and the degree of log^2(x) is 2, so the overall degree of the function is 2/3.

4. Can L'Hôpital's rule be used to solve this limit at infinity problem?

Yes, L'Hôpital's rule can be used to solve this limit at infinity problem if the resulting function after dividing by the highest power of x is of the form ∞/∞ or 0/0. In this case, the resulting function after simplification would be 0/∞, so L'Hôpital's rule can be applied to further simplify the function and evaluate the limit.

5. What is the final result of the limit at infinity for x^(2/3) / (log^2(x))?

After solving the limit using the steps mentioned above, the final result would be 0. This means that as x approaches infinity, the function x^(2/3) / (log^2(x)) approaches 0, or becomes infinitely small.

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