prove that this function differentiable endles times on x=0

??

 

 

if i will write the definition of the derivetive

i could prove that it differentiates on x=0

how to prove that for endless derivatives of this function

??

You can calculate the derivatives (to all orders) of exp(-1/x^2) using the chain rule and other basic calculus rules. Then you just need to check that these derivatives are equal to zero at x=0 (i.e. the derivative coming from the left equals the derivative coming from the right.) This is all that is needed to prove that derivatives of all orders exist.

if i will prove this for the first derivative

and the second derivative

and the third derivative

but its not prooving for the next endless derivatives

 

its only proving for these cases

??

i tried to prove for the first derivative

but i dont get a final limit here

[math]

f(x)=e^\frac{-1}{x^2} \\

[/math]

[math]

f'(0)=\lim_{x->0^+}\frac{e^\frac{-1}{x^2}-0}{x-0} \\

[/math]

[math]

f'(0)=\lim_{x->0^-}\frac{e^\frac{-1}{x^2}-0}{x-0}

[/math]

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the makloren formula is

[math]

f(x)=\sum_{n}^{k}\frac{f^{(k)}}{k!}x^k+o(x^n)

[/math]

 

how to take the n'th derivative from here??

 

where to put the formula in and get the n'th derivative??

 

this function

is for approximating so in order for me to get to the n'th members approximation

first i need to do manually one by one n times derivative

so its not helping me

??

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i was told

"once we can express the function as a power series around zero and it is differentiable at zero, we know it is infinitely differentiable"

differentiable around zero part:

[math]

f'(0)=\lim_{x->0}\frac{e^\frac{-1}{x^2}-0}{x-0} \\

=\lim_{x->0}\frac x{2e^{1/x^2}}

[/math]

i know that exponentials grow faster then polinomials but the power of the exp is 1/x^2

so the denominator goes to infinity but faster then the numenator so the expression

goes to 0.

power series around zero part:

the power series of e^x

[math]

g(x)=e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^3)\\

[/math]

i put -1/x^2 instead of x

[math]

g(x)=e^x=1+\frac{-1}{x^2}+\frac{x^{4}}{2}+\frac{-x^6}{6}+O(-x^6)\\

[/math]

 

what now??