limit of a $\ln(\cos(x))/x$ [closed]
Find the limit $$\lim \limits_{x \to 0} \dfrac{\ln(\cos x)}{x}$$ without L'Hôpital's rule.
I don't know how to find the limit.
Solution 1:
by the definition of the derivative, we know that
$f^`(a)=\lim_{x \to 0}\frac{f(x+a)-f(a)}{x}$
it can be observed that $f(0)=\ln(\cos(0))=0$ so according to definition
$f^`(0)=\lim_{x \to 0}\frac{\ln(\cos(x))-\ln(\cos(0))}{x}=\frac{-\sin(a)}{\cos(a)}|_{a=0}=0$