Question

$\underset{x\to \infty }{lim}\frac{x^5}{5^x}=$

• A. $0$
• B. $1$
• C. $\infty$
• D. does not exist

Answer 1

The correct option is A $0$

$\underset{x+\infty }{lim}\frac{x^5}{5^x}\left(\frac{\infty }{\infty }\right)$ from

By $L^1$-Hospital

$=\underset{x+\infty }{lim}\frac{5x^4}{5^{x}1x5}\frac{\infty }{\infty }$

Applying $L^{-1}$- Hospital four times

$\underset{x+\infty }{lim}\frac{5\times 4\times 3\times 2\times 1}{5^x(1x5{)}^5}$

$\underset{x+\infty }{lim}\frac{120}{5^x(1x5{)}^5}$

$=\frac{120}{\infty }=0$

$\therefore \underset{x+\infty }{lim}\frac{x^5}{5^x}=0$