Question

The value of $\displaystyle\lim_{x\to\infty} \left(\frac{x+5}{x+2}\right)^{x}$ is equal to

• A. $e^{-3}$
• B. $e$
• C. $e^{3}$
• D. $e^{2}$

Answer 1

Answer: (C)

$e^{3}$

Answer 2

We have,$\lim\limits_{x\to\infty}\left(\frac{x+5}{x+2}\right)^{x}$
$=\lim\limits_{x\to\infty}\left(1+\frac{3}{x+2}\right)^{x} (1^{\infty}$ form) $=e^{\lim\limits_{x\to\infty} \frac{3x}{2}} = e^{\lim\limits_{x\to\infty} \frac{3}{\left(1+2/ x\right)}} = e^{3}$