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''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>
The convergence can be improved:
<math>\exp(x)=\lim_{n \to \infty} \left(1 + \frac{x}{n + \frac{1}{6} \sin\left(\frac{\pi}{n}\right)}\right)^n</math>
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>
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