_ x 1 x^2-1+x-1 x^2-1 is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow 1} \frac{\sqrt{x^2-1}+\sqrt{x-1}}{\sqrt{x^2-1}}$ is equal to

A
$\frac{1}{2}$
B
$\sqrt{2}+1$
C
1
✓
$1+\frac{1}{\sqrt{2}}$

✓

Answer

Correct option: D.

$1+\frac{1}{\sqrt{2}}$

(D)
$\lim _{x \rightarrow 1} \frac{\sqrt{x^2-1}+\sqrt{x-1}}{\sqrt{x^2-1}}$
$=\lim _{x \rightarrow 1} \frac{\sqrt{(x-1)(x+1)}+\sqrt{x-1}}{\sqrt{(x-1)(x+1)}}$
$=\lim _{x \rightarrow 1} \frac{[\sqrt{x+1}+1]}{\sqrt{x+1}}$
$=\frac{\sqrt{2}+1}{\sqrt{2}}$
$=1+\frac{1}{\sqrt{2}}$

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