Using Napier’s inequality it is easy to show that:
.
So
From that the limit is clear.
Good afternoon,
I am working on a problem that asks to find the following limit:
This should be equal to e, but I am having some difficulty remembering why.
I checked WolframAlpha and was provided with the following:
Indeterminate form of type . Transform using
=
Where did this come from?
If you let Then take the log of both sides you can write:
Then take the limit of both sides.
Now use l-hospitals rule:
http://www.youtube.com/watch?v=6g_g9-AU188
See this thread
CB
Ah...
I think I follow this, however...
When we take the log of both sides and get:
Why are our new limits not:
Is the ln treated as a constant?
I hope my question is not too rediculous.
PS... I would love to watch your youtube video, but youtube is blocked at my place of work.
Firstly,
If we take the natural logarithms of both sides, we will be able to handle the "index" x before evaluating the limit using L'Hopital's rule.
This brings y into a form that can be evaluated as using L"Hopital's Rule, since if
evaluates to
Hence, differentiating numerator and denominator to apply L'Hopital's Rule..
Now this can be evaluated as x approaches infinity....
as
Therefore..
First off, thank you all for your continued efforts here.
Secondly, I apologize for being so slow to catch on here, requiring everyone to reiterate.
However, I think I have a more clear and concise understanding of where I'm losing you...
Right about here... how (or why, for that matter) did we end up with ?
Are we simply jostling the terms about to allow us to apply L'Hopital's rule?
I appologize if this process has been frustrating... it has been very taxing for me.