![Prove that: lim_x → ∞ [x + (1/x)]^x = e](/tpl/images/2491/7402/b91c6.jpg)
![\displaystyle\lim_{x \to \infty }\rm \bigg[1 + {\dfrac{1}{x} }\bigg]^{x} = e](/tpl/images/2491/7402/6f221.png)
![\rm :\longmapsto\:\displaystyle\lim_{x \to \infty }\rm \bigg[1 + {\dfrac{1}{x} }\bigg]^{x}](/tpl/images/2491/7402/80ed5.png)
![\rm :\longmapsto\:y = \displaystyle\lim_{x \to \infty }\rm \bigg[1 + {\dfrac{1}{x} }\bigg]^{x}](/tpl/images/2491/7402/b1316.png)
![\rm :\longmapsto\: log_{e}(y) = \displaystyle\lim_{x \to \infty }\rm log_{e} \bigg[1 + {\dfrac{1}{x} }\bigg]^{x}](/tpl/images/2491/7402/283b2.png)
![\rm :\longmapsto\: log_{e}(y) = \displaystyle\lim_{x \to \infty }\rm xlog_{e} \bigg[1 + {\dfrac{1}{x} }\bigg]](/tpl/images/2491/7402/32498.png)
![\rm :\longmapsto\: log_{e}(y) = \displaystyle\lim_{x \to \infty }\rm x \bigg[\dfrac{1}{x} - {\dfrac{1}{ {2x}^{2} } + \dfrac{1}{ {3x}^{3}} - \dfrac{1}{ {4x}^{4} } + - - - }\bigg]](/tpl/images/2491/7402/d1274.png)
![\rm :\longmapsto\: log_{e}(y) = \displaystyle\lim_{x \to \infty }\rm \bigg[1 - {\dfrac{1}{ 2x } + \dfrac{1}{ {3x}^{2}} - \dfrac{1}{ {4x}^{3} } + - - - }\bigg]](/tpl/images/2491/7402/019f1.png)
![\bf\implies \:\:\displaystyle\lim_{x \to \infty }\rm \bigg[1 + {\dfrac{1}{x} }\bigg]^{x} = e](/tpl/images/2491/7402/843d9.png)
Prove that: lim_x → ∞ [x + (1/x)]^x = e
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