I need help with:
Limit (cos3x)^5/x as x approaches 0
Should I use L'Hopital's Rule?
How do i begin to solve this?
I need help with the first few steps?
$\displaystyle \lim_{x \to 0} [\cos(3x)]^{\frac{5}{x}}$
let $\displaystyle y = [\cos(3x)]^{\frac{5}{x}}$
$\displaystyle \ln{y} = \frac{5}{x} \cdot \ln[\cos(3x)]$
$\displaystyle \lim_{x \to 0} \frac{5\ln[\cos(3x)]}{x}$
L'Hopital ...
$\displaystyle 5 \lim_{x \to 0} {\frac{-3\sin(3x)}{\cos(3x)}} = 5 \cdot 0 = 0$
$\displaystyle \lim_{x \to 0} \, \ln{y} = 0$
so ...
$\displaystyle \lim_{x \to 0} \, y = e^0 = 1$