Question

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

• A. $3$
• B. $5$
• C. $2$
• D. Any real number

Answer 1

Answer: (C)

$2$

GIven:${lim}_{x\to \infty }(\sqrt{x^4+{ax}^3+{3x}^2+bx+2}-\sqrt{x^4+2x-{cx}^2+2x-d})$

To find: value of a, if above limit is finite

Sol: On rationalising the limit we get,

${lim}_{x\to \infty }\frac{{(x}^4+a^3+3x^2+bx+2)-(x^4+{2x}^3-{cx}^2+2x-d)}{\{\sqrt{x^4+{ax}^3+{3x}^2+bx+2}+\sqrt{x^4+{2x}^3-{cx}^2+2x-d}\}}$

=${lim}_{x\to \infty }\frac{(a-2)x^3+(3-c)x^2+(b-2)x+(2-d)}{\sqrt{x^4+{ax}^3+{3x}^2+bx+2}+\sqrt{x^4+2x^3-{cx}^2+2x-d}}$

Since denominator is of order of $x^2$,

So the coefficient of $x^3$ is numerator has to be zero in the numerator $⟹a=2$