Proof: limit=0 for any positive integer n

[dincerekin]

Homework Statement

Prove that $$\lim_{x\to0}\frac{e^\frac{-1}{x^2}}{x^n}=0$$ for any positive integer n.

Homework Equations

The Attempt at a Solution

I've tried using a combination of induction and l'hopital's rule to no avail. Perhaps I am over complicating it?

All help is appreciated

Thanks in advance

 

[micromass]

Try to write it as

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

and apply l'Hopital now.

 

[dincerekin]

I applied l'hospital's and simplified a bit and now I've got

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

now what?

 

[micromass]

Simplify more and apply induction.

 

[hunt_mat]

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

 

[dincerekin]

so I am 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?

 

[Infinitum]

so I am 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?

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

 

[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 can't 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 $$\lim_{x\to0}{x^{-1}}$$ doesn't exist, right?

 

[micromass]

You did

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

But this is false.

 

[Infinitum]

so,

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

Uhh, this is wrong. Does multiplying x^-1 and x^1-k give x^2-k??

 

[dincerekin]

oh sorry, that should be the other way around

 

[dincerekin]

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

 

[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?

 

[Dick]

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

 

[dincerekin]

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?

oh! how didn't I see this before

thanks so much <3