- #1
Hernaner28
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Homework Statement
[tex]f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2}\sin \frac{1}{x}}&{x \ne 0}\\
0&{x = 0}
\end{array}} \right.[/tex]
Is it differentiable at x=0? If it is, what's its value?
Homework Equations
The Attempt at a Solution
I've calculated the derivative function for x not equal zero:
[tex]f'(x) = \left\{ {\begin{array}{*{20}{c}}
{2x\sin \frac{1}{x} - \cos \frac{1}{x}}&{x \ne 0}\\
0&{x = 0}
\end{array}} \right.[/tex]
And:
[tex]\mathop {\lim }\limits_{x \to 0} 2x\sin \frac{1}{x} - \cos \frac{1}{x}[/tex]
This limit doesn't exists so IT IS NOT DIFFERENTIABLE at 0. But if I use the definition of the incremental I DO get THAT IT IS differentiable at 0 and the derivative is 0. How could this be possible?Please, try to be the most clear as you can, I get easily confused with this things... Thanks!
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