lim-x-pi-4-sin-x-cos-x-tan-pi-8-x-2-

Question Number 175362 by cortano1 last updated on 28/Aug/22

\lim_{x \to \frac{π}{4}} \frac{\sin x - \cos x}{\tan (\frac{π}{8} - \frac{x}{2})} = ?

Commented by infinityaction last updated on 28/Aug/22

\lim_{x \to \frac{π}{4}} \frac{- \sqrt{2} \sin (π / 4 - x)}{\sin (\frac{π}{8} - \frac{x}{2}) \sec (\frac{π}{8} - \frac{x}{2})}

\lim_{x \to \frac{π}{4}} \frac{- 2 \sqrt{2} \sin (\frac{π}{8} - \frac{x}{2}) \cos (\frac{π}{8} - \frac{x}{2})}{\sin (\frac{π}{8} - \frac{x}{2})}

- 2 \sqrt{2}

Commented by CElcedricjunior last updated on 28/Aug/22

\lim_{x \to \frac{π}{4}} \frac{s i n x - c o s x}{t a n (\frac{π}{8} - \frac{x}{2})} = \frac{0}{0} = F I

e n d a p p l y h o s p i t a l

\lim_{x \to \frac{π}{4}} \frac{c o s x + s i n x}{- \frac{1}{2} [1 + t a \overset{2}{n} (\frac{π}{8} - \frac{x}{2})]} = \frac{\sqrt{2}}{- \frac{1}{2}}

\lim_{x \to \frac{π}{4}} \frac{s i n x - c o s x}{t a n (\frac{π}{8} - \frac{x}{2})} = - 2 \sqrt{2}

\dots \dots \dots l e c \overset{´}{e l} \overset{`}{e b r e} c e d r i c j u n i o r \dots \dots \dots

Answered by BaliramKumar last updated on 28/Aug/22

\lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} (s i n x - c o s x)}{\frac{d}{d x} t a n (\frac{π}{8} - \frac{x}{2})} = \lim_{x \to \frac{π}{4}} \frac{c o s x + s i n x}{- \frac{1}{2} {s e c}^{2} (\frac{π}{8} - \frac{x}{2})}

\frac{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}}{\frac{- 1}{2} \times 1} = - 2 \sqrt{2}