Proof: limit=0 for any positive integer n

1. May 12, 2012

dincerekin

1. The problem statement, all variables and given/known data
Prove that $$\lim_{x\to0}\frac{e^\frac{-1}{x^2}}{x^n}=0$$ for any positive integer n.

2. Relevant equations

3. The attempt at a solution
I've tried using a combination of induction and l'hopital's rule to no avail. Perhaps im over complicating it?

All help is appreciated

2. May 12, 2012

micromass

Try to write it as

$$\lim_{x\rightarrow 0} \frac{x^{-n}}{e^{1/x^2}}$$

and apply l'Hopital now.

3. May 12, 2012

dincerekin

I applied l'hopitals and simplified a bit and now ive got

$$\frac{n}{2}\lim_{x\to0}\frac{x^2}{e^\frac{1}{x^2}x^n}=0$$

now what?

4. May 12, 2012

micromass

Simplify more and apply induction.

5. May 12, 2012

hunt_mat

Write $e^{-\frac{1}{x^{2}}}$ as a power series and then perform the limiting process.

6. May 12, 2012

dincerekin

so im trying to show its true for n=k+1 assuming n=k

i.e i need to show that $$\lim_{x\to0}\frac{x^{1-k}}{e^\frac{1}{x^2}}=0$$

assuming that $$\lim_{x\to0}\frac{x^{2-k}}{e^\frac{1}{x^2}}=0$$

im not sure how to manipulate this now?

7. May 12, 2012

Infinitum

Split the first term into the one you know(assumed), and its factor. Then apply basic limits.

8. May 12, 2012

dincerekin

so,

$$\lim_{x\to0}\frac{x^{2-k}}{e^\frac{1}{x^2}}=\lim_{x\to0}\frac{x^{-1}x^{1-k}}{e^\frac{1}{x^2}}$$

but i cant simply say that
$$\lim_{x\to0}\frac{x^{-1}x^{1-k}}{e^\frac{1}{x^2}}= \lim_{x\to0}{x^{-1}} × \lim_{x\to0}\frac{x^{1-k}}{e^\frac{1}{x^2}}$$

because $$\lim_{x\to0}{x^{-1}}$$ doesnt exist, right?

9. May 12, 2012

micromass

You did

$$x^{2-k}=x^{-1}x^{1-k}$$

But this is false.

10. May 12, 2012

Infinitum

Uhh, this is wrong. Does multiplying x-1 and x1-k give x2-k??

11. May 12, 2012

dincerekin

oh sorry, that should be the other way around

12. May 12, 2012

dincerekin

it should be
$$\lim_{x\to0}\frac{x^{1-k}}{e^\frac{1}{x^2}}=\lim_{x\to0}\frac{x^{-1}x^{2-k}}{e^\frac{1}{x^2}}$$

13. May 12, 2012

micromass

Uuuh, or slightly more useful

$$\lim_{x\rightarrow 0}\frac{x^{2-k}}{e^{1/x^2}}=\lim_{x\rightarrow 0}\frac{x\cdot x^{1-k}}{e^{1/x^2}}$$

Can you split up the limits now?

14. May 12, 2012

Dick

I would take the absolute value and then look at the log of your expression.

15. May 12, 2012

dincerekin

oh! how didn't I see this before

thanks so much <3