_ x x^2+a^2-x^2+b^2 x^2+c^2-x^2+d^2 = - Vidyadip

MCQ

$\lim _{x \rightarrow\infty} \frac{\sqrt{x^2+a^2}-\sqrt{x^2+b^2}}{\sqrt{x^2+c^2}-\sqrt{x^2+d^2}}=$

✓
$\frac{a^2-b^2}{c^2-d^2}$
B
$\frac{a^2+b^2}{c^2-d^2}$
C
$\frac{a^2+b^2}{c^2+d^2}$
D
$\frac{a^2-b^2}{c^2+d^2}$

✓

Answer

Correct option: A.

$\frac{a^2-b^2}{c^2-d^2}$

(A)
$\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+ a ^2}-\sqrt{x^2+ b ^2}}{\sqrt{x^2+ c ^2}-\sqrt{x^2+ d ^2}}$
$=\lim _{x \rightarrow \infty} \frac{\left( a ^2- b ^2\right)}{\left( c ^2- d ^2\right)}\left[\frac{\sqrt{1+\frac{ c ^2}{x^2}}+\sqrt{1+\frac{ d ^2}{x^2}}}{\sqrt{1+\frac{ a ^2}{x^2}+\sqrt{1+\frac{ b ^2}{x^2}}}}\right]$
$=\frac{ a ^2- b ^2}{ c ^2- d ^2}$

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