Calculate-lim-x-5-x-8-x-1-3x-

Question Number 162618 by LEKOUMA last updated on 30/Dec/21

C a l c u l a t e

\lim_{x \to \infty} {(5^{x} + 8^{x})}^{\frac{1}{3 x}}

Answered by Ar Brandon last updated on 30/Dec/21

A = \lim_{x \to \infty} {(5^{x} + 8^{x})}^{\frac{1}{3 x}}

\ln A = \lim_{x \to \infty} \frac{1}{3 x} \ln (5^{x} + 8^{x})

\ln A = \lim_{x \to \infty} \frac{1}{3 x} \ln [8^{x} (1 + \frac{5^{x}}{8^{x}})]

\ln A = \lim_{x \to \infty} {\frac{1}{3 x} {\ln 8}^{x} + \frac{1}{3 x} \ln (1 + {(\frac{5}{8})}^{x})}

\ln A = \lim_{x \to \infty} {\frac{1}{3} \ln 8 + \frac{1}{3 x} {(\frac{5}{8})}^{x}} = \frac{1}{3} \ln 8 = \ln 2

A = 2

Answered by tounghoungko last updated on 31/Dec/21

\lim_{x \to \infty} {[8^{x} ({(\frac{5}{8})}^{x} + 1)]}^{\frac{1}{3 x}} =

\lim_{x \to \infty} 2 {({(\frac{5}{8})}^{x} + 1)}^{\frac{1}{3 x}} = 2 \times 1 = 2