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Consider the following function $f:\Bbb R^2\to \Bbb R$: $$f(x,y) = \begin{cases} \frac{1}{x^2 + y^2} \cdot e^{-\frac{1}{\sqrt{x^2 + y^2}}}, & \text{if $(x,y) \neq (0,0)$} \\[2ex]0, & \text{if $(x,y) = (0,0)$} \end{cases}$$

Determine if $\pmb f$ differentiable at $\pmb{(0,0)}$.

I started by checking if the partial derivatives exist at $(0,0)$: $$f_x(0,0)=\lim\limits_{h \to 0} \frac{f(h+0,0)-f(0,0)}{h}=\lim\limits_{h \to 0} \frac{\frac{1}{h^2}\cdot e^{-\frac{1}{{h}}}}{h}=\lim\limits_{h \to 0} \frac{e^{-\frac{1}{{h}}}}{h^3}$$

I'm stuck here as I'm not sure how to solve the above limit. Please advise.

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1 Answer 1

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$\sqrt {h^{2}}=|h|$ so you should have $e^{-1/|h|}$ instead of $e^{-1/h}$. Put $t=\frac 1 {|h|}$ and use the fact that $\lim_{t\to \infty} e^{-t} t^{a}=0$ for every real number $a$.

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  • $\begingroup$ Thanks for the explanation. $\endgroup$
    – roman asa
    Jun 12, 2019 at 0:54

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