Question

The value of ${lim}_{x\to -\infty }\frac{x^5\cos \left(\frac{1}{\pi x^2}\right)+x^6\sin \left(\frac{1}{\pi x}\right)+7}{|x{|}^5+6|x|+7}$ is

• A. $(\pi +1)$
• B. $-(1+\pi )$
• C. $-(1+\frac{1}{\pi })$
• D. $-(1+\frac{2}{\pi })$

Answer 1

Answer: (C)

$-(1+\frac{1}{\pi })$

$\begin{array}{c}Wehave\\ {lim}_{x\to -\infty }\frac{x^5\cos \left(\frac{1}{\pi x^2}\right)+x^6\sin \left(\frac{1}{\pi x}\right)+7}{{\left|x\right|}^5+6\left|x\right|+7}\\ Applyingansionsweget,\\ {lim}_{x\to -\infty }\frac{x^5\left\{1-\frac{{\left(\frac{1}{\pi x^2}\right)}^2}{2!}+\frac{{\left(\frac{1}{\pi x^2}\right)}^4}{4!}-\frac{{\left(\frac{1}{\pi x^2}\right)}^6}{6!}\right\}+x^6\left\{1-\frac{1}{\pi x}-\frac{{\left(\frac{1}{\pi x}\right)}^3}{3!}+\frac{{\left(\frac{1}{\pi x}\right)}^5}{5!}....\infty \right\}+7}{\left|x^5\right|+6\left|x\right|+7}\\ {lim}_{x\to -\infty }\frac{x^5\left[\left(1+\frac{1}{\pi }\right)+\frac{1}{x^4}\left(-\frac{1}{{\pi }^{2}2!}+\frac{1}{{\pi }^{5}5!}\right)\right]-\frac{x^3}{{\pi }^{3}3!}+\mathrm{7....}\infty }{\left|x^5\right|+6\left|x\right|+7}\\ {lim}_{x\to -\infty }\frac{x^5\left[\left(1+\frac{1}{\pi }\right)+\frac{1}{x^4}\left(-\frac{1}{{\pi }^{2}2!}+\frac{1}{{\pi }^{5}5!}\right)\right]-\frac{x^3}{{\pi }^{3}3!}+\mathrm{7....}\infty }{{\left|x\right|}^5\left(1+\frac{6}{{\left|x\right|}^4}+\frac{7}{{\left|x\right|}^5}\right)}\\ Applyingthelimit,weget\\ -\left(1+\frac{1}{\pi }\right).\\ Hence,theoptionCisthecorrectanswer.\end{array}$