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Math Help - Definition of e in limit

  1. #1
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    Definition of e in limit

    How do I prove that:
    \lim h \to 0 \left ( 1+h+\frac{h^{2}}{2} \right )^{\frac{1}{h}}=e

    I know that the definition of e is:
    \lim n \to \infty \left ( 1+\frac{1}{n} \right )^{n} =e
    Do I compare these two limits to get the answer?
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  2. #2
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    Re: Definition of e in limit

    Quote Originally Posted by bilalsaeedkhan View Post
    How do I prove that:
    \lim h \to 0 \left ( 1+h+\frac{h^{2}}{2} \right )^{\frac{1}{h}}=e

    I know that the definition of e is:
    \lim n \to \infty \left ( 1+\frac{1}{n} \right )^{n} =e
    Do I compare these two limits to get the answer?
    \displaystyle \lim_{h \to 0}\left(1 + h + \frac{h^2}{2}\right)^{\frac{1}{h}} &= \lim_{h \to 0}e^{\ln{\left(1 + h + \frac{h^2}{2}\right)^{\frac{1}{h}}}} \\ &= \lim_{h \to 0}e^{\frac{\ln{\left(1 + h + \frac{h^2}{2}\right)}}{h}} \\ &= e^{\lim_{h \to 0}\frac{\ln{\left(1 + h + \frac{h^2}{2}\right)}}{h}} \\ &= e^{\lim_{h \to 0}\frac{\frac{1 + h}{1 + h + \frac{h^2}{2}}}{1}} \textrm{ by L'Hospital's Rule} \\ &= e^{\lim_{h \to 0}\frac{1 + h}{1 + h + \frac{h^2}{2}}}  \\ &= e^{\frac{1 + 0}{1 + 0 + \frac{0^2}{2}}} \\ &= e^1 \\ &= e
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