calculate-lim-x-0-cos-x-cot-x-

Question Number 188819 by mnjuly1970 last updated on 07/Mar/23

calculate

\lim_{x \to 0^{+}} {(\sqrt{\cos (\sqrt{x})})}^{\cot (x)} = ?

Commented by mehdee42 last updated on 07/Mar/23

calculate

\lim_{x \to 0^{+}} {(\sqrt{\cos (\sqrt{x})})}^{\cot (x)} = ?

n o t i c

i f {l i m}_{x \to a} f (x) = 1 & {l i m}_{x \to a} g (x) = \infty

\Rightarrow {l i m}_{x \to a} f {(x)}^{g (x)} = e^{{l i m}_{x \to a} (f (x) - 1) g (x)}

{l i m}_{x \to 0^{+}} \frac{\sqrt{c o s \sqrt{x}} - 1}{t a n x} \overset{h o p}{=} {l i m}_{x \to a} \frac{\frac{- s i n \sqrt{x}}{2 \sqrt{x}}}{2 (1 + t a n x) \sqrt{c o s \sqrt{x}}} = - \frac{1}{4}

\Rightarrow a n s w r i s : \frac{1}{\sqrt[4]{e}}

Answered by mahdipoor last updated on 07/Mar/23

y = {(\sqrt{c o s (\sqrt{x})})}^{c o t (x)} = {[c o s (\sqrt{x})]}^{\frac{c o t (x)}{2}}

\Rightarrow l n (y) = \frac{c o t (x)}{2} l n (c o s (\sqrt{x})) = \frac{l n (c o s (\sqrt{x}))}{2 t a n (x)}

\Rightarrow x \to 0^{+} l i m [l n (y)] = l i m \frac{l n (c o s \sqrt{x})}{2 t a n (x)}

\Rightarrow= \frac{0}{0}, H o p \Rightarrow= \frac{\frac{- s i n (\sqrt{x})}{c o s (\sqrt{x})} . \frac{1}{2 \sqrt{x}}}{\frac{2}{{c o s}^{2} (x)}} =

\frac{- {c o s}^{2} (x)}{4 c o s (\sqrt{x})} \times [\frac{s i n (\sqrt{x})}{\sqrt{x}}] = \frac{- 1}{4} \times \frac{s i n \sqrt{x}}{\sqrt{x}}

\Rightarrow h o p \Rightarrow \frac{c o s (\sqrt{x}) \times \frac{1}{2 \sqrt{x}}}{\frac{1}{2 \sqrt{x}}} = c o s \sqrt{x} = 1

\Rightarrow\Rightarrow l i m l n (y) = \frac{- 1}{4} \times 1 \Rightarrow l i m y = \frac{1}{^{4} \sqrt{e}}

\dots \dots \dots \dots \dots N o t e :

g e t l i m x \to c \frac{f (x)}{g (x)} =, f (c) = g (c) = 0

\frac{f (x)}{g (x)} = k (x) \frac{u (x)}{v (x)} t h a t k (c) \neq 0, u (c) = v (c) = 0

\Rightarrow h o p \Rightarrow \frac{k^{^{'}} (c) u (c) + k (c) u^{^{'}} (c)}{v^{^{'}} (c)} = k (c) \frac{u^{^{'}} (c)}{v^{^{'}} (c)}

\dots \dots \dots \dots \dots

Commented by mnjuly1970 last updated on 07/Mar/23

s e p a s o s t a d

Answered by qaz last updated on 07/Mar/23

\underset{x \to 0^{+}}{l i m} {(\sqrt{\cos \sqrt{x}})}^{\cot x} = \underset{x \to 0^{+}}{l i m} {(1 - \frac{1}{4} x + o (x))}^{\frac{1}{x + o (x)}} = e^{- 4}

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