Limit involving exponential function

In summary: I haven't solved this either, but want to post my thoughts until someone else can help. I did the same thing you did to begin.Let L= \lim_{x \rightarrow \infty}\left( \frac{x}{1+x^2} \right)^x\ln(L) = \lim_{x \rightarrow \infty}x \ln\left( \frac{x}{1+x^2} \right) = \lim_{x \rightarrow \infty} x \left[ \ln(x)-\ln(1+x^2) \right]= \lim_{x \rightarrow \infty} x\ln(
  • #1
Bueno
13
0
Hello everyone, how are you?

I'm having trouble to evalue the following limit:

\(\displaystyle \lim_{x->\infty} (\frac{x}{1+x^2})^x \)

I "transformed" it into \(\displaystyle e^{ln{(\frac{x}{1+x^2})^x}}\) and tried to solve this limit:

\(\displaystyle \lim_{x->\infty} x ln{(\frac{x}{1+x^2})}\)

But I have no idea how to solve it correctly. Can you help me?

Thank you,

Bueno
 
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  • #2
I haven't solved this either, but want to post my thoughts until someone else can help. I did the same thing you did to begin.

Let \(\displaystyle L= \lim_{x \rightarrow \infty}\left( \frac{x}{1+x^2} \right)^x\)

\(\displaystyle \ln(L) = \lim_{x \rightarrow \infty}x \ln\left( \frac{x}{1+x^2} \right) = \lim_{x \rightarrow \infty} x \left[ \ln(x)-\ln(1+x^2) \right]= \lim_{x \rightarrow \infty} x\ln(x)-x\ln(1+x^2)\)

This seems good so far and appropriate now to use L'Hopital's Rule, but I see one problem. The answer to this limit is $0$, so if $L=0$ then $\ln(L)$ isn't defined and I don't see how we can get the answer this way, so it seems like another method should be used. This limit uses a trick I haven't seen before or forgot, because it's tougher than many I have been browsing through just now.

Will post back if I see the solution and I hope that someone helps you soon! :)
 
  • #3
We may rewrite the expression as:

\(\displaystyle \left(\frac{\frac{1}{x}}{1+\frac{1}{x^2}} \right)^x\)

and we see we have the form:

\(\displaystyle 0^{\infty}\)

which is not indeterminate, and is in fact equal to zero.
 
  • #4
Nicely done, Mark. (Clapping). I knew that the indeterminate form method wasn't going to work and thought of rewriting the expression somehow but just didn't see it.
 
  • #5
I couldn't see anything like this, it worked perfectly. It's the first limit I see that needs this kind of trick.
Thank you!

I've been studying this kind of limits today and most of them were solved by the technique I mentioned in the first post, except for the one you just showed me how to solve and another.

Bueno
 
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FAQ: Limit involving exponential function

What is a limit involving exponential function?

A limit involving exponential function is a mathematical concept that describes the behavior of a function as the input approaches a specific value. In this case, the function involves an exponential function, which is a function in the form f(x) = a^x, where a is a constant and x is the input.

How do you evaluate a limit involving exponential function?

To evaluate a limit involving exponential function, you can use the rules of limits or L'Hôpital's rule, which allows you to take the derivative of both the numerator and denominator to find the limit. You can also use properties of exponential functions, such as the fact that the limit of a^x as x approaches a is a^a.

What is the relationship between limits and exponential functions?

Limits and exponential functions are closely related, as exponential functions are often used to model real-world situations where quantities grow or decay at a constant rate. Limits involving exponential functions can help us understand the behavior of these functions and make predictions about their values at specific points.

Can a limit involving exponential function be undefined?

Yes, a limit involving exponential function can be undefined if the function approaches infinity or negative infinity as the input value approaches a specific value. This can occur when the base of the exponential function is greater than 1, or when the input value is a negative number.

What are some real-world applications of limits involving exponential function?

Limits involving exponential function can be used in various fields, such as finance, biology, and physics. In finance, they can be used to calculate compound interest or the growth of investments. In biology, they can be used to model population growth. In physics, they can be used to describe the behavior of radioactive decay or other natural phenomena.

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