Having trouble with a limit

Hi,

I'm stuck on finding this limt: $\lim_{x\rightarrow 0}\frac{x^{n}e^{x}}{1-e^{x}}$.

I've tried what I think to be every approach (L'hopital's, L'hopitals with induction, sandwich rule, manipulating the function etc.) but have had no success.

Does anyone know how to go about solving this?

Thanks,
Dan

definitely L hospital for me

(and let the purists argue...)

Hi,

I'm stuck on finding this limt: $\lim_{x\rightarrow 0}\frac{x^{n}e^{x}}{1-e^{x}}$.

I've tried what I think to be every approach (L'hopital's, L'hopitals with induction, sandwich rule, manipulating the function etc.) but have had no success.

Does anyone know how to go about solving this?

Thanks,
Dan

Could you expand a little...? haha

I think l'Hopital will work - too late in the evening for me to try this though

One thing to note to help yourself out a bit - note that the e^x in the numerator is irrelevant: because it tends to a finite non-zero limit as x->0 it doesn't affect whether or not the whole thing tends to a limit so you can "factor it out" and multiply it back in using the algebra of limits at the end.

You haven't said what n is, however - is it a general integer, a natural number or what? This will affect your conclusion!

EDIT: I've just thought of another way of doing it too, which is to write the e^x in the numerator in terms of the denominator and then separating things out.

I think l'Hopital will work - too late in the evening for me to try this though

One thing to note to help yourself out a bit - note that the e^x in the numerator is irrelevant: because it tends to a finite non-zero limit as x->0 it doesn't affect whether or not the whole thing tends to a limit so you can "factor it out" and multiply it back in using the algebra of limits at the end.

You haven't said what n is, however - is it a general integer, a natural number or what? This will affect your conclusion!

EDIT: I've just thought of another way of doing it too, which is to write the e^x in the numerator in terms of the denominator and then separating things out.

sorry but I was teaching late tonight and I typed L'hospital without much thinking, so this is what I would do

write exponentials (top and bottom) as power series

on the top the series will start with x^n plus higher order terms
on the denominator, once the 1s cancell out it will start with -x plus higher order terms

divide an x all the way through should be enough to show that the limit is in fact zero

PS I have assumed that n is positive integer/positive real

Thanks for your comment. n is a positive integer I couldn't get this to work but I have a tutorial session soon and will ask about this way of doing it.



Ahhhhh! Very clever . Thanks very much. I wouldn't have thought of that!

Hi,

I'm stuck on finding this limt: $\lim_{x\rightarrow 0}\frac{x^{n}e^{x}}{1-e^{x}}$.

I've tried what I think to be every approach (L'hopital's, L'hopitals with induction, sandwich rule, manipulating the function etc.) but have had no success.

Does anyone know how to go about solving this?

Thanks,
Dan

Thanks for your comment. n is a positive integer I couldn't get this to work but I have a tutorial session soon and will ask about this way of doing it.



Ahhhhh! Very clever . Thanks very much. I wouldn't have thought of that!