Question

Find the limits of $\frac{\sqrt{a^2+ax+x^2}-\sqrt{a^2-ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}},$ when $x=0$.

Answer 1

To find the limits of $L=\frac{\sqrt{a^2+ax+x^2}-\sqrt{a^2-ax+x^{{}^2}}}{\sqrt{a+x}-\sqrt{a-x}}$ when $x=0$

Using Rationalisation, we get

$L=\underset{x\to 0}{lim}\frac{\sqrt{a^2+ax+x^2}-\sqrt{a^2-ax+x^{{}^2}}}{\sqrt{a+x}-\sqrt{a-x}}\times \frac{\sqrt{a^2+ax+x^2}+\sqrt{a^2-ax+x^{{}^2}}}{\sqrt{a^2+x^2+ax}+\sqrt{a^2-ax+x^2}}$

$=\underset{x\to 0}{lim}\frac{(a^2+ax+x^2)-(a^2-ax+x^2)}{(\sqrt{a+x}-\sqrt{a-x})\times (\sqrt{a^2+ax+x^2}+\sqrt{a^2-ax+x^2})}$

$=\underset{x\to 0}{lim}\frac{2ax}{(\sqrt{a+x}-\sqrt{a-x})\times (\sqrt{a^2+ax+x^2}+\sqrt{a^2-ax+x^2})}$

Again using rationalisation, we get

$L=\underset{x\to 0}{lim}\frac{2ax}{(\sqrt{a+x}-\sqrt{a-x})\times (\sqrt{a^2+ax+x^2}+\sqrt{a^2-ax+x^2})}\times \frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}$

$=\underset{x\to 0}{lim}\frac{2ax\times (\sqrt{a+x}+\sqrt{a-x})}{\left(2x\right)\times (\sqrt{a^2+ax+x^2}+\sqrt{a^2-ax+x^2})}$

$=\underset{x\to 0}{lim}\frac{a\times (\sqrt{a+x}+\sqrt{a-x})}{(\sqrt{a^2+ax+x^2}+\sqrt{a^2-ax+x^2})}$

$=\frac{a\times (\sqrt{a+0}+\sqrt{a-0})}{(\sqrt{a^2+a\left(0\right)+0^2}+\sqrt{a^2-a\left(0\right)+0^2})}$

$=\frac{a\times 2\sqrt{a}}{2a}$

$=\sqrt{a}$