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,
$\underset{x\to \infty }{\lim }\left(\sqrt{x^4+a{x}^3+3x^2+bx+2}-\sqrt{x^4+2x^3-c{x}^2+3x-d}\right)$
$=\underset{x\to \infty }{\lim }\frac{\left(x^4+a{x}^3+3x^2+bx+2\right)-\left(x^4+2x^3-c{x}^2+3x-d\right)}{\sqrt{x^4+a{x}^3+3x^2+bx+2}+\sqrt{x^4+2x^3-c{x}^2+3x-d}}$
$=\underset{x\to \infty }{\lim }\frac{(a-2)x^3+(3+c)x^2+(b-3)x+(2+d)}{\sqrt{x^4+a{x}^3+3x^2+bx+2}+\sqrt{x^4+2x^3-c{x}^2+3x-d}}$
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, $a-2=0$
$\Rightarrow a=2$