lim-x-pi-2-cos-x-sin-x-cos-x-1-3-sin-x-

Question Number 161319 by cortano last updated on 16/Dec/21

\lim_{x \to \frac{π}{2}} \frac{\cos x}{\sqrt[3]{\sin x + \cos x} - \sin x} = ?

Answered by som(math1967) last updated on 16/Dec/21

\underset{x \to \frac{π}{2}}{l i m} \frac{c o s x {{(s i n x + c o s x)}^{\frac{2}{3}} + {(s i n x + c o s x)}^{\frac{1}{3}} s i n x + {s i n}^{2} x}}{s i n x + c o s x - {s i n}^{3} x}

\underset{x \to \frac{π}{2}}{l i m} \frac{c o s x {{(s i n x + c o s x)}^{\frac{2}{3}} + {(s i n x + c o s x)}^{\frac{1}{3}} s i n x + {s i n}^{2} x}}{s i n x (1 - {s i n}^{2} x) + c o s x}

\underset{x \to \frac{π}{2}}{l i m} \frac{c o s x {{(s i n x + c o s x)}^{\frac{2}{3}} + {(s i n x + c o s x)}^{\frac{1}{3}} s i n x + {s i n}^{2} x}}{c o s x (s i n x c o s x + 1)}

\underset{x \to \frac{π}{2}}{l i m} \frac{{{(s i n x + c o s x)}^{\frac{2}{3}} + {(s i n x + c o s x)}^{\frac{1}{3}} s i n x + {s i n}^{2} x}}{s i n x c o s x + 1}

\frac{1 + 1 + 1}{0 + 1} = 3