eMath-lim-x-pi-sin-x-pi-tan-x-pi-tan-x-

Question Number 109091 by bemath last updated on 21/Aug/20

\frac{△ ♭ e M a t h ▽}{\equiv⊸\equiv⊸\equiv}

\lim_{x \to π} \frac{\sin x}{\sqrt{π + \tan x} - \sqrt{π - \tan x}} ?

Answered by bobhans last updated on 21/Aug/20

Commented by bobhans last updated on 21/Aug/20

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

\lim_{x \to π} \frac{\sin x . \cos x}{2 \sin x} \times 2 \sqrt{π} = - \sqrt{π}

Answered by bemath last updated on 21/Aug/20

s e t x = π + q

\lim_{q \to 0} \frac{- \sin q}{\sqrt{π + \tan q} - \sqrt{π - \tan q}} =

\lim_{q \to 0} \frac{- \sin q}{\sqrt{π} {\sqrt{1 + \frac{\tan q}{π}} - \sqrt{1 - \frac{\tan q}{π}}}} =

\frac{1}{\sqrt{π}} . \lim_{q \to 0} \frac{- q}{(1 + \frac{q}{2 π}) - (1 - \frac{q}{2 π})} =

\frac{1}{\sqrt{π}} . \lim_{q \to 0} \frac{- q}{(\frac{q}{π})} = \frac{- π}{\sqrt{π}} = - \sqrt{π} .

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