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DM
Numerade Educator

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Problem 59 Medium Difficulty

Determine whether $ f'(0) $ exists.

$ f(x) = \left\{
\begin{array}{ll}
x \sin \frac{1}{x} & \mbox{if $ x \neq 0 $}\\
0 & \mbox{if $ x = 0 $}
\end{array} \right.$

Answer

$f(0)$ does not exist

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Video Transcript

So in this problem were given this function F of X equals X Sign of one over X. If x is not equal to zero and zero if X equals zero, we were asked to determine F prime and zero. Well, IF F Promise zero exists, then The limit as X approaches zero. Uh F of X -F of zero Over X -0 0 exists. So let's now look at this. Okay, So this is the limit as X goes to zero of X, Sign of one over X minus zero, Because F0 is zero over X. So this is the limit as X goes to zero Of sign of one over X. Okay, well, the problem is As X goes to zero when over X goes to infinity and sign of infinity, oscillates in the interval from -1-1 and does not converge. So by that then the limit as X goes to zero, Sign of one over X does not exist. So F prime zero does not exist.