Question

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

Answer 1

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

$\left[\frac{\infty }{\infty }form\right]$

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

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

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

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