Question

$\displaystyle \lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}$ is equal to

• A. $e^{4}$
• B. $e^{6}$
• C. $e^{5}$
• D. $e$

Answer 1

Answer: (C)

$\ \displaystyle\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}= \displaystyle\lim _{x \rightarrow \infty}\left(1+\frac{5}{x+1}\right)^{x+4}$
$=e^{5 \displaystyle\lim _{x \rightarrow \infty}\left(\frac{x+4}{x+1}\right)}=x^{5 \displaystyle\lim _{x \rightarrow \infty}\left(\frac{1+\frac{4}{x}}{1+\frac{1}{x}}\right)}$
$=e^{5}$