Question

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

• A. 1
• B. -1
• C. 0
• D. 1/2

Answer 1

The correct option is C

0

${lim}_{x\to \infty }\frac{x^5}{5^x}={lim}_{x\to \infty }\frac{x^5}{e^{xlog5}}={lim}_{x\to \infty }\frac{x^5}{e^{kx}},\text{where k=log5}={lim}_{x\to \infty }\frac{x^5}{(1+kx+\frac{k^2{x}^2}{2!}+\frac{k^3{x}^3}{3!}+\frac{k^4{x}^4}{4!}+\frac{k^5{x}^5}{5!}+\frac{k^6{x}^6}{6!}+......)}={lim}_{x\to \infty }\frac{1}{[\frac{1}{x^5}+k.\frac{1}{x^4}+\frac{k^2}{2!}.\frac{1}{x^3}+\frac{k^3}{3!}.\frac{1}{x^2}+\frac{k^4}{4!}.\frac{1}{x^1}+\frac{k^5}{5}+(\frac{k^6}{6!}x+......)]}=0$ As denominator is approaching to infinity.