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Thread: Definition of exponential functions

  1. #1
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    Definition of exponential functions

    I do understand that $\displaystyle e$ is the number such that
    $\displaystyle e=\lim_{n \rightarrow \infty} \left( 1+\frac{1}{n} \right)^n$
    and how it is obtained.

    But I don't understand how the definition of the function $\displaystyle e^x$ as a limit of a sequence is obtained, i.e.

    $\displaystyle e^x=\lim_{n\rightarrow\infty}\left( 1+\frac{x}{n} \right)^n$

    Is there a proof?

    Appreciate those who help. Thanks!
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  2. #2
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    We can expand it by Binomial Theorem

    $\displaystyle ( 1+ \frac{x}{n} )^n = 1 + (n) \frac{x}{n} + \frac{n(n-1)}{2} (\frac{x}{n})^2 + .... $

    By taking the limit , n tends to infinity ,

    $\displaystyle e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + .... $
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  3. #3
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    great, thanks!
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