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Calculus / Calculus: Early Transcendentals 1 / Chapter 9.4 / Problem 20E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Chapter 8, Problem 20E

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QUESTION:

Limits Evaluate the following limits using Taylor series.

\(\lim \limits_{x \rightarrow 0} \frac{(1-2 x)^{-1 / 2}-e^{x}}{8 x^{2}}\)

Questions & Answers

QUESTION:

Limits Evaluate the following limits using Taylor series.

\(\lim \limits_{x \rightarrow 0} \frac{(1-2 x)^{-1 / 2}-e^{x}}{8 x^{2}}\)

ANSWER:

Solution 20E

Step 1:

In this problem we have to evaluate $\lim\limits_{{x} \rightarrow {0}}\ \frac{(1-2x{)}^{-1/2}\ -\ {e}^{x}}{8{x}^{2}}$ using Taylor series.

Consider $\lim\limits_{{x} \rightarrow {0}}\ \frac{(1-2x{)}^{-1/2}\ -\ {e}^{x}}{8{x}^{2}}$

We know the Taylor series of $(1-2x{)}^{-1/2}=1+x+\frac{3}{2}{x}^{2}+\frac{5}{2}{x}^{3}+\frac{35}{8}{x}^{4}+...$

The Taylor series of ${e}^{x}\ =1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}+...$

Conside

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