Question

The value of ${lim}_{x\to \infty }{\left(\frac{x^5}{5^x}\right)}^{}$ is

• A. $1$
• B. $-1$
• C. $0$
• D. none of these

Answer 1

The correct option is C $0$

It is of form $\frac{\infty }{\infty }$, hence applying l'opital,

$\underset{x\to \infty }{lim}\frac{5x^4}{5^{x}log5}$, again of form $\frac{\infty }{\infty }$,

again by l'optial, $\underset{x\to \infty }{lim}\frac{5.4x^3}{5^x(log5{)}^2}$, similarly

we would get $\frac{\mathrm{5.4.3.2.1}}{5^x(log5{)}^5}$ which tends to as $x\to \infty$ (as $5^x\to \infty$)

$\Rightarrow \left(C\right)$.