Question

The value of $\displaystyle\lim _{x \rightarrow \infty} \frac{3^{x+1}-5^{x+1}}{3^{x}-5^{x}}$ is

• A. $5$
• B. $\frac{1}{5}$
• C. $-5$
• D. None of these

Answer 1

Answer: (A)

$\displaystyle\lim _{x \rightarrow \infty} \frac{3^{x+1}-5^{x+1}}{3^{x}-5^{x}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{3 \cdot 3^{x}-5 \cdot 5^{x}}{3^{x}-5^{x}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{3 \cdot\left(\frac{3}{5}\right)^{x}-5}{\left(\frac{3}{5}\right)^{x}-1}=\frac{-5}{-1}$
$=5$
$\left(\because \displaystyle\lim _{n \rightarrow \infty} a^{n}=0\right.$, if $\left.-1< a< 1\right)$