# Homework Help: 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