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Question
\[\lim_{x \to \infty} \frac{\sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}}{\sqrt{x^2 + c^2} - \sqrt{x^2 + d^2}}\]
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Solution
\[\lim_{x \to \infty} \left[ \frac{\sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}}{\sqrt{x^2 + c^2} - \sqrt{x^2 + d^2}} \right]\]
\[\text{ Rationalising the numerator and the denominator }:\]
\[ \lim_{x \to \infty} \left[ \frac{\left( \sqrt{x^2 + a^2} - \sqrt{x^2 + b^2} \right)}{\left( \sqrt{x^2 + c^2} - \sqrt{x^2 + d^2} \right)} \times \frac{\left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right)}{\left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right)} \times \frac{\left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right)}{\left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right)} \right]\]
\[ = \lim_{x \to \infty} \left[ \frac{\left( \sqrt{x^2 + a^2} - \sqrt{x^2 + b^2} \right) \left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right) \left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right)}{\left( \sqrt{x^2 + c^2} - \sqrt{x^2 + d^2} \right) \left( \sqrt{x^2 + c^2} + \sqrt{x^2 + d^2} \right) \left( \sqrt{x^2 + a^2} + \sqrt{x^2 + b^2} \right)} \right]\]
\[ = \lim_{x \to \infty} \frac{\left( x^2 + a^2 \right) - \left( x^2 + b^2 \right)}{\left( x^2 + c^2 \right) - \left( x^2 + d^2 \right)} \times \left( \frac{\sqrt{x^2 + c^2} + \sqrt{x^2 + d^2}}{\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2}} \right)\]
\[ {= \lim}_{x \to \infty} \left( \frac{a^2 - b^2}{c^2 - d^2} \right) \left( \frac{\sqrt{x^2 + c^2} + \sqrt{x^2 + d^2}}{\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2}} \right)\]
\[\text{ Dividing the numerator and the denominator byx }:\]
\[ \lim_{x \to \infty} \left( \frac{a^2 - b^2}{c^2 - d^2} \right) \left( \frac{\sqrt{1 + \frac{c^2}{x^2}} + \sqrt{1 + \frac{d^2}{x^2}}}{\sqrt{1 + \frac{1}{x^2}} + \sqrt{1 + \frac{b^2}{x^2}}} \right)\]
\[As x \to \infty , \frac{1}{x}, \frac{1}{x^2} \to 0\]
\[ = \left( \frac{a^2 - b^2}{c^2 - d^2} \right) \left( \frac{\sqrt{1} + \sqrt{1}}{\sqrt{1} + \sqrt{1}} \right)\]
\[ = \frac{a^2 - b^2}{c^2 - d^2}\]
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