A limit positive integer and real number

[quasar987]

[SOLVED] A limit

Homework Statement

How do you show that

$$\lim_{x\rightarrow a}\frac{e^{-a^2/(a^2-x^2)}}{(a^2-x^2)^{2m}(x-a)}=0$$

for 'm' a positive integer and 'a' a real number >0??This is a type 0/0 indeterminate form but l'Hospital's rule is not helpful because when you differentiate the denominator, you make the degree 4m+1 polynomial of the denominator drop 1 degree, but you make a (-2xa²)/(a²-x²)² appear in the numerator.

And Mapple says "undefined" when I plug a=3

 

[StatusX]

Try a change of variables.

 

[quasar987]

It seems I tried every change of variable possible but none help... :/

 

[StatusX]

How about taking your new variable to be what's in the exponential. You'll be left with something like:

$$\lim_{u \rightarrow \infty} e^{-u} p(u) f(u)$$

where p(u) is a polynomial, and f(u) is something that looks like a polynomial for large u. It shouldn't be too hard from here.

 

[quasar987]

Got it. I had made an error in calculating.

Thanks StatusX.