Math Doubts

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


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


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.

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

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


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.

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more