I was asked a question for homework to find the derivative of the following function:
$$f(x) =\begin{cases} x^3(1+\sin \left(\frac{1}{x})\right)&\text{if}\; x \neq 0\\ \\ 0&\text{if}\;x=0 \end{cases}$$
I have started off by stating epsilon-delta definition of a limit by stating that for all $\varepsilon >0$, there exists a $\delta >0$ such that if $$0<|x-a|< \delta, \;\text{then}\; |f(x) - L|< \varepsilon$$
I have then stated that since the limit of the function is $0$ (By first principles) then $$\lim_{h \to0} \frac{f(0+h)-f(0)}{h} =\lim_{h \to 0} \frac{h^3(1+\sin (1/h))}{h}$$
This then can simplify down and by substituting the function into the definition for when the limit is $0$ we get, $|h^2(1+\sin(1/x)) - 0|< \varepsilon$.
And this is where I get stuck. I'm not too sure how to find the solution from this point onwards. Any help would be greatly appreciated. Thanks in advance.