Question

The value of $\displaystyle \lim_{x \to \infty}$ $\sqrt{a^{2}x^{2}+ax+1}-\sqrt{a^{2}x^{2}+1}$ is

• A. $\frac{1}{2}$
• B. $1$
• C. $2$
• D. None of these

Answer 1

Answer: (A)

Given $\displaystyle \lim_{x \to \infty}$ $\sqrt{a^{2}x^{2}+ax+1}-\sqrt{a^{2}x^{2}+1}$
$=\displaystyle \lim_{x \to \infty}$ $\frac{a^{2}x^{2}+ax+1-a^{2}x^{2}-1}{\sqrt{a^{2}x^{2}+ax+1}+\sqrt{a^{2}x^{2}+1}}$
$=\displaystyle \lim_{x \to \infty}$ $\frac{ax}{x\left[\sqrt{a^{2}+\frac{a}{x}+\frac{1}{x^{2}}}+\sqrt{a^{2}+\frac{1}{x^{2}}}\right]}$
$=\frac{a}{\sqrt{a^{2}}+\sqrt{a^{2}}}=\frac{a}{2a}$
$=\frac{1}{2}$