Question

$\lim_{x\to\infty} \left[ \sqrt{x^{2} + ax +b } -x \right] \left(a < 0 < b \right)$

• A. depends on both $a$ and $b$
• B. depends only on $b$
• C. depends only on $a$
• D. does not depend on $a$ and $b$

Answer 1

Answer: (C)

Given that, $\displaystyle\lim _{x \rightarrow \infty}\left[\sqrt{x^{2}+a x+b}-x\right]$
$=\displaystyle\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+a x+b}-x\right) \times\left(\frac{\sqrt{x^{2}+a x+b}+x}{\sqrt{x^{2}+a x+b}+x}\right)$
[By rationalisation]
$=\displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}+a x+b-x^{2}}{\sqrt{x^{2}+a x+b}+x}=\displaystyle\lim _{x \rightarrow \infty} \frac{a x+b}{\sqrt{x^{2}+a x+b}+x}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{x\left(a+\frac{b}{x}\right)}{x\left(\sqrt{ \left.1+\frac{a}{x}+\frac{b}{x^{2}}+1\right)}\right.}$
(taking $x$ common from both numerator and denominator)
$ = \frac{a}{(1 + 1)} = \frac{a}{2}$
It is clear that it depends only on $a$