(a) State the limit definition of the derivative of a function f at the point x = a.
(b) Use (a) to find f'(0).
f(x)={x^2cos(ln|x|) if x =/= 0
{0 if x=0
I'm pretty sure you use the squeeze theorem
The limit definition of $\displaystyle f^{'}(x)$ is
$\displaystyle f^{'}(x)=\lim_{h \rightarrow 0}\Bigg{(} \frac{(x+h)^{2}\cos(ln|x+h|)-x^2\cos(ln|x|)}{h} \Bigg{)} $
Now for a=0, this means
$\displaystyle f^{'}(0)=\lim_{h \rightarrow 0} \Bigg{(} \frac{h^2\cos(ln|h|)}{h} \Bigg{)} $ (*)
But since $\displaystyle -1<\cos(x)<1 , \forall x$
$\displaystyle \lim_{h \rightarrow 0} -h < (*) < \lim_{h \rightarrow 0} h $
Apply squeeze theorem to conclude that f('0) = 0.