[latex]\lim_{x \to \infty} ( \frac{3x}{3x+2})^{x+5}[/latex]

[latex] \lim_{x \to \infty} ( \frac{3x}{3x+2})^x^+^5= \lim_{x \to \infty} 3^x^+^5( \frac{x}{3x+2})^x^+^5= \\ \lim_{x \to \infty} exp(log(3)(x+5)+log( \frac{x}{3x+2})(x+5))= \\ exp( \lim_{x \to \infty} (log(3)(x+5)+log( \frac{x}{3x+2})(x+5))= \\ exp( \lim_{x \to \infty} log(3^x^+^5exp((x+5)(log(x)-log(3x+2))))=e^-^ \frac{2}{3}= \frac{1}{e^ \frac{2}{3} } [/latex]