# $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize {\Big(1+\dfrac{1}{x}\Big)}^x}$ formula

## Formula

$\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize {\Big(1+\dfrac{1}{x}\Big)}^x}$ $\,=\,$ $e$

### Introduction

The limit of $1+\dfrac{1}{x}$ raised to the power of $x$ as $x$ approaches infinity is equal to mathematical constant $e$.

In limits, the exponential functions similar to this function are often appeared. So, it’s considered as a standard result and used as a formula in calculus for evaluating the limits of exponential functions when its input tends to infinity.

#### Other form

This standard result of limits can also be written in terms of any variable.

$(1) \,\,\,$ $\displaystyle \large \lim_{g \,\to\, \infty}{\normalsize {\Big(1+\dfrac{1}{g}\Big)}^g}$ $\,=\,$ $e$

$(2) \,\,\,$ $\displaystyle \large \lim_{y \,\to\, \infty}{\normalsize {\Big(1+\dfrac{1}{y}\Big)}^y}$ $\,=\,$ $e$

### Proof

Learn how to prove the limit of $x$-th power of binomial $1+\dfrac{1}{x}$ as $x$ approaches infinity is equal to $e$ in mathematics.

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