Evaluate $LaTeX: \displaystyle \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{\frac{5 x}{8}}$

This is an indeterminate form of the type $LaTeX: \displaystyle 1^\infty$ . Taking the natural logarithm of both sides gives: $LaTeX: \ln(L) = \ln\left( \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{\frac{5 x}{8}} \right)$ Pulling the limit out of the continuous function and using log properties gives: $LaTeX: \ln(L) = \lim_{x \to \infty}\frac{5 x}{8}\ln\left(1 + \frac{2}{x} \right)$ This is an indeterminate form of the type $LaTeX: \displaystyle 0 \cdot \infty$ . Converting it to type $LaTeX: \displaystyle \frac{0}{0}$ and using L'Hospitials rule gives: $LaTeX: \ln(L) = \lim_{x \to \infty}\frac{\ln\left(1 + \frac{2}{x}\right)}{\frac{8}{5 x}} = \frac{- \frac{2}{x^{2}}}{- \frac{8}{5 x^{2}}} = \frac{5}{4}$ Solving for $LaTeX: \displaystyle L$ gives $LaTeX: \displaystyle L = e^{\frac{5}{4}}$