Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s). lim_{xrightarrow0}frac{e^{ax}-1}{x}

Trent Carpenter 2020-11-09 Answered
Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
limx0eax1x
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joshyoung05M
Answered 2020-11-10 Author has 97 answers
The given limit limx0eax1x can be evaluated as,
The Taylor series expansion of eax is,
eax=1+ax1!+(ax)22!+(ax)33+...
Therefore,
=limx0(eax1x)=limx0((1+ax1!+(ax)22!+(ax)33+...)1x)
=limx0(a+a2x2!+a3x23!+...)
=a
Hence, limx0eax1x=a
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Jeffrey Jordon
Answered 2022-04-01 Author has 2087 answers

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