Question

Use L'Hoptals rule:
lim x->0 (cos(x))^(1/x^2)
I have no idea where to start since this isn't a fraction

Answer 1

make it into a fraction by using logs

Answer 2

As @Zarkon said
\[
\ln \left(\sqrt[x^2]{\cos
(x)}\right)=\frac{\ln (\cos (x))}{x^2}
\]

Answer 3

if x->0 we have 0/0 so you can apply L'Hopital's rule

Answer 4

Take the derivative of the numerator over the derivative of the denominator, you get after simplification
\[
-\frac{\sin (x)}{(2 x) \cos (x)}\to -\frac 1 2
\]
When x goes to zero.
It is easy now to find the original limit

Answer 5

Thank you

Answer 6

YW. Do you know how to finish it?

Answer 7

Yeah I at first got e^0.5 but then your work showed me that Dx(cosx) was -sinx/cosx not sinx/cosx. So then I got e^-0.5

Answer 8

Yes, the answer is
\[
\frac{1}{\sqrt{e}}
\]

Answer 9

Thank you I knew how to start it after Zarkon's reply but I give eliassaab the best response because of my error taking the derivative that he corrected