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$\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}$ equals
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Verified Answer
The correct answer is:
$\mathrm{e}^{3}$
$$
\begin{array}{l}
\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}=\lim _{x \rightarrow \infty}\left(1+\frac{3}{x+2}\right)^{x+3} \\
=\lim _{x \rightarrow \infty}\left[\left(1+\frac{3}{x+2}\right)^{\frac{x+2}{3}}\right]^{\frac{3(x+3)}{x+2}} \\
=e^{\left.\lim _{x \rightarrow \infty}(3] \frac{1+\frac{3}{x}}{x+\frac{2}{x}}\right)}=e^{3}
\end{array}
$$
\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}
$$
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