"...are not indeterminate forms. Find the following by inspection:"
$\displaystyle\lim_{ x\to \pi/2} (\cos x)^{\tan x}$
and
$\displaystyle\lim _{x\to \pi/2} [ (2/\pi-2x) + \tan x ]$
These are from a L'hopitals worksheet, so I have no clue how to do these. Does anyone know how I can at least start? Does the first one require ' ln ' ?
Thank you.
For first , of the form $0^{\infty}$. $$\lim_{x \to \pi/2}(\cos x)^{\tan x}=\lim_{x \to \pi/2}\left(e^{\log(\cos x)}\right)^{\tan x}=\lim_{x \to \pi/2}\exp\left(\tan x \log(\cos x)\right)$$Now see that, when $x\to \pi/2$ then $\tan x\to \infty$ and $\log(\cos x)\to -\infty$. So product tends to $-\infty$. Hence , required limit is $\exp(-\infty)=0$