nice-calculus-I-lim-x-0-cos-x-1-3-cot-x-solution-sin-x-x-x-0-

Question Number 139949 by mnjuly1970 last updated on 02/May/21

\dots \dots \dots n i c e \dots . * \dots . * \dots . * \dots . c a l c u l u s (I)

ϕ = {l i m}_{x \to 0^{+}} {(\sqrt[3]{c o s (\sqrt{x}})}^{c o t (x)} = ? ?

s o l u t i o n \dots \dots \dots

s i n (x) \approx x (x \to 0)

ϕ := {l i m}_{x \to 0^{+}} {{(c o s (\sqrt{x}))}^{\frac{1}{3}}}^{c o t (x)} = {l i m}_{x \to 0^{+}} {{(1 - \frac{x}{2})}^{c o t (x)}}^{\frac{1}{3}}

:= {l i m}_{x \to 0^{+}} {{(1 - \frac{x}{2})}^{\frac{1}{s i n (x) \approx x}}}^{\frac{c o s (x)}{3}} \underset{c o s (x) \to 1 (x \to 0)}{\overset{⟨ {l i m}_{x \to 0} {(1 - a x)}^{\frac{1}{x}} = \frac{1}{e^{a}} ⟩}{=}} e^{\frac{- 1}{6}} = \sqrt[6]{\frac{1}{e}} \dots .