Q70 / 120 · VITEEE 2009 · Mathematics
$\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}$ equals
- A$\mathrm{e}$
- B$\mathrm{e}^{2}$
- Correct answerC$\mathrm{e}^{3}$
- D$\mathrm{e}^{5}$
Solution
$$
\begin{array}{l}
\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}=\lim _{x \rightarrow \infty}\left(1+\frac{3}{x+2}\right)^{x+3} \\
=\lim _{x \rightarrow \infty}\left[\left(1+\frac{3}{x+2}\right)^{\frac{x+2}{3}}\right]^{\frac{3(x+3)}{x+2}} \\
=e^{\left.\lim _{x \rightarrow \infty}(3] \frac{1+\frac{3}{x}}{x+\frac{2}{x}}\right)}=e^{3}
\end{array}
$$
Hence, the correct answer is option: 'C'.