Question

If $$ \lim _{x \rightarrow \infty}\left\{\left(\sqrt{x^{4}+a x^{3}+3 x^{2}+b x+2}-\sqrt{x^{4}+2 x^{3}-c x^{2}+3 x-d}\right)\right\} $$ is finite, then the value of a is

• A. 3
• B. 5
• C. 2
• D. any real number

Answer 1

Answer: (C)

We have,
$\displaystyle\lim _{x \rightarrow \infty}\left(\sqrt{x^{4}+a x^{3}+3 x^{2}+b x+2}-\sqrt{x^{4}+2 x^{3}-c x^{2}+3 x-d}\right)$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{\left(x^{4}+a x^{3}+3 x^{2}+b x+2\right)-\left(x^{4}+2 x^{3}-c x^{2}+3 x-d\right)}{\sqrt{x^{4}+a x^{3}+3 x^{2}+b x+2}+\sqrt{x^{4}+2 x^{3}-c x^{2}+3 x-d}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{(a-2) x^{3}+(3+c) x^{2}+(b-3) x+(2+d)}{\sqrt{x^{4}+a x^{3}+3 x^{2}+b x+2}+\sqrt{x^{4}+2 x^{3}-c x^{2}+3 x-d}}$
Clearly, the degree of the polynomial in numerator is $3$ and that of denominator is $2$. Therefore, for the limit to be finite, we must have, $a-2=0 $
$\Rightarrow a=2$