(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

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- Feb 17th 2010, 10:30 AMryno16Find a derivative using squeeze theorem
(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 - Feb 17th 2010, 11:36 AMrunning-gag
Hi

The cosine of any expression is always between -1 and 1 - Feb 17th 2010, 12:02 PMvince

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.