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Rationalization Method to Remove Indeterminate Form

lim x →∞√x2+a...

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

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

$[\frac{\infty}{\infty} f o r m]$

By Rationalising,

${lim}_{x \to \infty} \frac{(\sqrt{x^{2} + a^{2}} - \sqrt{x^{2} + b^{2}})}{(\sqrt{x^{2} + c^{2}} - \sqrt{x^{2} + d^{2}})} \times \frac{(\sqrt{x^{2} + a^{2}} + \sqrt{x^{2} + b^{2}})}{(\sqrt{x^{2} + a^{2}} + \sqrt{x^{2} + b^{2}})}$

${lim}_{x \to \infty} \frac{(x^{2} + a^{2}) - (x^{2} - b^{2})}{(\sqrt{x^{2} + c^{2}} - \sqrt{x^{2} + d^{2}}) (\sqrt{x^{2} + a^{2}} + \sqrt{x^{2} + b^{2}})}$

${lim}_{x \to \infty} \frac{(a^{2} - b^{2})}{(\sqrt{x^{2} + c^{2}} - \sqrt{x^{2} + d^{2}}) (\sqrt{x^{2} + a^{2}} + \sqrt{x^{2} + b^{2}})}$

${lim}_{x \to \infty} \frac{(a^{2} - b^{2}) (\sqrt{x^{2} + c^{2}} + \sqrt{x^{2} + d^{2}})}{(\sqrt{x^{2} + c^{2}} - \sqrt{x^{2} + d^{2}}) (\sqrt{x^{2} + c^{2}} + \sqrt{x^{2} + d^{2}}) (\sqrt{x^{2} + a^{2}} + \sqrt{x^{2} + b^{2}})}$

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

Divide the numerator and denominator by x

${lim}_{x \to \infty} \frac{(a^{2} - b^{2})}{(c^{2} - d^{2})} ⎡ ⎢ ⎢ ⎣ \frac{\sqrt{1 + \frac{c^{2}}{x^{2}}} + \sqrt{1 + \frac{d^{2}}{x^{2}}}}{\sqrt{1 + \frac{1}{x^{2}}} + \sqrt{1 + \frac{b^{2}}{x^{2}}}} ⎤ ⎥ ⎥ ⎦$

$A s x \to \infty, \frac{1}{x}, \frac{1}{x^{2}} \to 0$

$= (\frac{a^{2} - b^{2}}{c^{2} - d^{2}}) (\frac{\sqrt{1} + \sqrt{1}}{\sqrt{1} + \sqrt{1}}) = \frac{a^{2} - b^{2}}{c^{2} - d^{2}}$

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Similar questions

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}}}

\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}}}

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1, \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1, - \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

\in R^{+}

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

- \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

x + i y = \sqrt{\frac{a + i b}{c + i d}}

{(x^{2} + y^{2})}^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

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