lim-x-pi-2-cosx-cotx-

Question Number 144544 by mohammad17 last updated on 26/Jun/21

{l i m}_{x \to \frac{π}{2}} {(c o s x)}^{c o t x}

Commented by mohammad17 last updated on 26/Jun/21

? ? ? ? ? ?

Answered by Olaf_Thorendsen last updated on 26/Jun/21

f (x) = {(\cos x)}^{\cot x}

g (x) = f (\frac{π}{2} - x) = {(\sin x)}^{\tan x}

\ln g (x) = \tan x . \ln (\sin x)

\ln g (x) \underset{0}{\sim} x \ln (x) \to 0

\Rightarrow g (x) \underset{0}{\to} 1

\lim_{x \to \frac{π}{2}} f (x) = \lim_{x \to 0} g (x) = 1

Answered by mathmax by abdo last updated on 26/Jun/21

f (x) = {(cosx)}^{cotanx} \Rightarrow f (x) = e^{ctanxlog (cosx)}

=_{\frac{π}{2} - x = t} e^{cotan (\frac{π}{2} - t) \log (sinx)} = e^{tant \log (sint)}

we have tant \log (sint) = sintlog (sint) \times \frac{1}{cost} \sim t logt \to 0 \Rightarrow

\lim_{x \to \frac{π}{2}} f (x) = e^{0} = 1