JS-lim-x-1-x-2-1-1-x-x-1-

Question Number 106883 by john santu last updated on 07/Aug/20

◊ JS ⧫

\lim_{x \to - 1} \frac{\sqrt{x^{2} - 1} + 1 - \sqrt{- x}}{\sqrt{x + 1}} ?

Answered by bemath last updated on 07/Aug/20

@ bemath @

\lim_{x \to - 1} \frac{\sqrt{x + 1} \sqrt{x - 1} + 1 - \sqrt{- x}}{\sqrt{x + 1}} =

\lim_{x \to - 1} \sqrt{x - 1} + \frac{1 - \sqrt{- x}}{\sqrt{x + 1}} \times \frac{1 + \sqrt{- x}}{1 + \sqrt{- x}} =

\lim_{x \to - 1} \sqrt{x - 1} + \frac{1 + x}{\sqrt{x + 1} (1 + \sqrt{- x})} =

\lim_{x \to - 1} (\sqrt{x - 1} + \frac{\sqrt{x + 1}}{1 + \sqrt{- x}}) = i \sqrt{2} ?

Answered by Dwaipayan Shikari last updated on 07/Aug/20

\lim_{x \to - 1} \sqrt{x - 1} + \frac{1 - \sqrt{- x}}{\sqrt{x + 1}}

\lim_{x \to - 1} i \sqrt{2} + \frac{1 + x}{\sqrt{x + 1}} . \frac{1}{1 + \sqrt{x}} = \lim_{x \to - 1} i \sqrt{2} + \frac{\sqrt{x + 1}}{1 + \sqrt{x}} = i \sqrt{2}