Solved: "Use Taylor series to evaluate the following limits...."

Trent Carpenter

Answered question

2020-11-09

Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).

lim_{x \to 0} \frac{e^{a x} - 1}{x}

joshyoung05M

Skilled2020-11-10Added 97 answers

The given limit

lim_{x \to 0} \frac{e^{a x} - 1}{x}

can be evaluated as,
The Taylor series expansion of

e^{a x}

is,

e^{a x} = 1 + \frac{a x}{1!} + \frac{(a x)^{2}}{2!} + \frac{(a x)^{3}}{3} + . . .

Therefore,

= lim_{x \to 0} (\frac{e^{a x} - 1}{x}) = lim_{x \to 0} (\frac{(1 + \frac{a x}{1!} + \frac{(a x)^{2}}{2!} + \frac{(a x)^{3}}{3} + . . .) - 1}{x})

= lim_{x \to 0} (a + \frac{a^{2} x}{2!} + \frac{a^{3} x^{2}}{3!} + . . .)

= a

Hence,

lim_{x \to 0} \frac{e^{a x} - 1}{x} = a

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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