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$\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+a^2}-\sqrt{x^2+b^2}}{\sqrt{x^2+c^2}-\sqrt{x^2+d^2}}=$
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Verified Answer
The correct answer is:
$\frac{a^2-b^2}{c^2-d^2}$
\(\lim _{x \rightarrow \infty}\left[\frac{\sqrt{x^2+a^2}-\sqrt{x^2+b^2}}{\sqrt{x^2+c^2}-\sqrt{x^2+d^2}}\right]\)
Rationalising the numerator and the denominator :
\(\begin{aligned}
& \lim _{x \rightarrow \infty}\left[\frac{\left(\sqrt{x^2+a^2}-\sqrt{x^2+b^2}\right)}{\left(\sqrt{x^2+c^2}-\sqrt{x^2+d^2}\right)} \times \frac{\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)}{\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)} \times \frac{\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)}{\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)}\right] \\
& =\lim _{x \rightarrow \infty}\left[\frac{\left(\sqrt{x^2+a^2}-\sqrt{x^2+b^2}\right)\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)}{\left(\sqrt{x^2+c^2}-\sqrt{x^2+d^2}\right)\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)}\right] \\
& =\lim _{x \rightarrow \infty} \frac{\left(x^2+a^2\right)-\left(x^2+b^2\right)}{\left(x^2+c^2\right)-\left(x^2+d^2\right)} \times\left(\frac{\sqrt{x^2+c^2}+\sqrt{x^2+d^2}}{\sqrt{x^2+a^2}+\sqrt{x^2+b^2}}\right) \\
& =\lim _{x \rightarrow \infty}\left(\frac{a^2-b^2}{c^2-d^2}\right)\left(\frac{\sqrt{x^2+c^2}+\sqrt{x^2+d^2}}{\sqrt{x^2+a^2}+\sqrt{x^2+b^2}}\right)
\end{aligned}\)
Dividing the numerator and the denominator by x :
\(\begin{aligned}
& \lim _{x \rightarrow \infty}\left(\frac{a^2-b^2}{c^2-d^2}\right)\left(\frac{\sqrt{1+\frac{c^2}{x^2}}+\sqrt{1+\frac{d^2}{x^2}}}{\sqrt{1+\frac{1}{x^2}}+\sqrt{1+\frac{b^2}{x^2}}}\right) \\
& A s x \rightarrow \infty, \frac{1}{x}, \frac{1}{x^2} \rightarrow 0 \\
& =\left(\frac{a^2-b^2}{c^2-d^2}\right)\left(\frac{\sqrt{1}+\sqrt{1}}{\sqrt{1}+\sqrt{1}}\right) \\
& =\frac{a^2-b^2}{c^2-d^2}
\end{aligned}\)
Rationalising the numerator and the denominator :
\(\begin{aligned}
& \lim _{x \rightarrow \infty}\left[\frac{\left(\sqrt{x^2+a^2}-\sqrt{x^2+b^2}\right)}{\left(\sqrt{x^2+c^2}-\sqrt{x^2+d^2}\right)} \times \frac{\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)}{\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)} \times \frac{\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)}{\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)}\right] \\
& =\lim _{x \rightarrow \infty}\left[\frac{\left(\sqrt{x^2+a^2}-\sqrt{x^2+b^2}\right)\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)}{\left(\sqrt{x^2+c^2}-\sqrt{x^2+d^2}\right)\left(\sqrt{x^2+c^2}+\sqrt{x^2+d^2}\right)\left(\sqrt{x^2+a^2}+\sqrt{x^2+b^2}\right)}\right] \\
& =\lim _{x \rightarrow \infty} \frac{\left(x^2+a^2\right)-\left(x^2+b^2\right)}{\left(x^2+c^2\right)-\left(x^2+d^2\right)} \times\left(\frac{\sqrt{x^2+c^2}+\sqrt{x^2+d^2}}{\sqrt{x^2+a^2}+\sqrt{x^2+b^2}}\right) \\
& =\lim _{x \rightarrow \infty}\left(\frac{a^2-b^2}{c^2-d^2}\right)\left(\frac{\sqrt{x^2+c^2}+\sqrt{x^2+d^2}}{\sqrt{x^2+a^2}+\sqrt{x^2+b^2}}\right)
\end{aligned}\)
Dividing the numerator and the denominator by x :
\(\begin{aligned}
& \lim _{x \rightarrow \infty}\left(\frac{a^2-b^2}{c^2-d^2}\right)\left(\frac{\sqrt{1+\frac{c^2}{x^2}}+\sqrt{1+\frac{d^2}{x^2}}}{\sqrt{1+\frac{1}{x^2}}+\sqrt{1+\frac{b^2}{x^2}}}\right) \\
& A s x \rightarrow \infty, \frac{1}{x}, \frac{1}{x^2} \rightarrow 0 \\
& =\left(\frac{a^2-b^2}{c^2-d^2}\right)\left(\frac{\sqrt{1}+\sqrt{1}}{\sqrt{1}+\sqrt{1}}\right) \\
& =\frac{a^2-b^2}{c^2-d^2}
\end{aligned}\)
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