# Limits of derivatives of an exponential

1. Oct 30, 2012

### Catria

1. The problem statement, all variables and given/known data

Determine the lowest derivative order for which the limit towards 0+ of the nth order derivative of f is nonzero (or otherwise does not exist). f = $e^{\frac{-1}{x^{2}}}$

2. Relevant equations

$lim_{x\rightarrow0+}\frac{d^{n}}{dx^{n}}e^{\frac{-1}{x^{2}}}$

3. The attempt at a solution

$lim_{x\rightarrow0+}\frac{d}{dx}e^{\frac{-1}{x^{2}}}$ = 0

$lim_{x\rightarrow0+}\frac{d^{2}}{dx^{2}}e^{\frac{-1}{x^{2}}}$ = 0

$lim_{x\rightarrow0+}\frac{d^{3}}{dx^{3}}e^{\frac{-1}{x^{2}}}$ = 0

2. Oct 31, 2012

### SammyS

Staff Emeritus
Use the chain rule.

$\displaystyle \frac{d}{dx}e^{-1/x^2}= \frac{2e^{-1/x^2}}{x^3}\ .$

Last edited: Oct 31, 2012
3. Oct 31, 2012

### Catria

I tried that at the first three orders and I still had the limit of these derivatives towards 0+ as 0.

4. Oct 31, 2012

### SammyS

Staff Emeritus
Right!

I get zero for the fourth derivative also.

I don't see how it will ever be anything else, no matter how high the order of the derivative, but I haven't proved that to myself.

.

5. Oct 31, 2012

### HallsofIvy

Staff Emeritus
Any derivative is $e^{-1/x^2}$ over a polynomial and its limit as x goes to 0 will always be 0.