Q103 / 160 · TS EAMCET 2009 · Mathematics

$\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}$ equals

  • A
    $e$
  • B
    $e^2$
  • Correct answer
    C
    $e^3$
  • D
    $e^5$

Solution

$\begin{gathered}\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}=\lim _{x \rightarrow \infty}\left(1+\frac{3}{x+2}\right)^{x+3} \\ =\lim _{x \rightarrow \infty}\left[\left(1+\frac{3}{x+2}\right)^{\frac{x+2}{3}}\right]^{\frac{3(x+3)}{x+2}}\end{gathered}$ $=e^{\lim _{x \rightarrow \infty} 3\left(\frac{1+\frac{3}{x}}{1+\frac{2}{x}}\right)}=e^3$

Hence, the correct answer is option: 'C'.

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