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

MCQ

$\mathop {\lim }\limits_{x \to \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
None of these

✓

Answer

Correct option: A.

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

a
$(a)$ $\mathop {\lim }\limits_{x \to \infty } \,\frac{{({a^2} - {b^2})}}{{({c^2} - {d^2})}}\,\frac{{\left[ {\sqrt {1 + \frac{{{c^2}}}{{{x^2}}}} + \sqrt {1 + \frac{{{d^2}}}{{{x^2}}}} } \right]}}{{\left[ {\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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