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Limit involving exponential function

Bueno

New member
Mar 29, 2013
14
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
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,043
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! :)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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.
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,043
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.
 

Bueno

New member
Mar 29, 2013
14
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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