[latex] \lim_{x \to \infty} ( \frac{x+3}{x+2} )^ x^{+1} [/latex]

[latex] \lim_{ x \to \infty } { ( \frac{x+3}{x+2} )^{x+1} } = \lim_{ x \to \infty } { ( \frac{x+2+1}{x+2} )^{x+2-1} = \lim_{ x \to \infty } { ( \frac{x+2}{x+2} + \frac{1}{x+2} )^{ (x+2) - 1 } = [/latex] [latex] = \lim_{ x \to \infty } { ( 1 + \frac{1}{x+2} )^{x+2} \cdot \lim_{ x \to \infty } { ( 1 + \frac{1}{x+2} )^{-1} = [/latex] [latex] = \lim_{ y = x+2 \to \infty } { ( 1 + \frac{1}{y} )^y \cdot \lim_{ y = x+2 \to \infty } { \frac{1}{ 1 + 1/y } = [/latex] [latex] = e \cdot \frac{1}{ 1 + 0 } = e \cdot 1 = e . [/latex]