Exponential function: Difference between revisions

Revision as of 21:14, 14 May 2025

Revision as of 21:19, 14 May 2025

Line 77:

===Limit of integer powers===

''The exponential function is the [[limit (mathematics)|limit]], as the integer {{mvar|n}} goes to infinity,<ref name="Maor"/><ref name=":0" />

<math display=block>\exp(x)=\lim_{n \to +\infty} \left(1+\frac xn\right)^n,.</math>

where <math>n</math> takes only integer values (otherwise, the exponentiation would require the exponential function to be defined). By continuity of the logarithm, this can be proved by taking logarithms and proving

<math display=block>x=\lim_{n\to\infty}\ln \left(1+\frac xn\right)^n= \lim_{n\to\infty}n\ln \left(1+\frac xn\right),</math>

for example with [[Taylor's theorem]].