Math Doubts

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

Formula

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

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.

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

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

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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