--| home | research | txt | img | code | teach | specialfunctionswiki | timescalewiki | hyperspacewiki | links |--
Back to the class

Quiz 6
Let $f(x) = \left\{ \begin{array}{ll} x \sin \left( \dfrac{1}{x} \right), & \quad x >0 \\ 0, & \quad x \leq 0 \end{array} \right.$
Does $f'(0)$ exist? If so, what is it? If not, why not?
Solution: $f'(0)$ does not exist. Try to calculate it: $$\begin{array}{ll} f'(0) &\stackrel{\mathrm{def}}{=} \displaystyle\lim_{t \rightarrow 0} \dfrac{f(t)-f(0)}{t-0} \\ &=\displaystyle\lim_{t \rightarrow 0} \dfrac{t \sin \left( \frac{1}{t} \right)}{t} \\ &=\displaystyle\lim_{t \rightarrow 0} \sin \left( \dfrac{1}{t} \right), \end{array}$$ but this limit does not exist (it oscillates wildly between $-1$ and $1$ faster and faster as $t$ approaches zero).