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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.4.66
Chapter 11, Problem 11.4.66

Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
lim ₓ→₀ (eᵃˣ − 1)/x

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Identify the limit expression: \(\lim_{x \to 0} \frac{e^{a x} - 1}{x}\), where \(a\) is a nonzero real parameter.
Recall the Taylor series expansion of the exponential function around \(x=0\): \(e^{a x} = 1 + a x + \frac{(a x)^2}{2!} + \frac{(a x)^3}{3!} + \cdots\).
Substitute the Taylor series expansion into the limit expression: \(\frac{e^{a x} - 1}{x} = \frac{\left(1 + a x + \frac{(a x)^2}{2!} + \cdots \right) - 1}{x} = \frac{a x + \frac{(a x)^2}{2!} + \cdots}{x}\).
Simplify the fraction by dividing each term in the numerator by \(x\): \(\frac{a x}{x} + \frac{(a x)^2}{2! x} + \cdots = a + \frac{a^2 x}{2!} + \cdots\).
Evaluate the limit as \(x \to 0\): since all terms involving \(x\) vanish, the limit simplifies to \(a\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits and Continuity

Limits describe the behavior of a function as the input approaches a particular value. Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms like 0/0, is essential for analyzing function behavior near specific points.
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Intro to Continuity

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. Using the Taylor series around zero (Maclaurin series) helps approximate functions near that point, simplifying limit evaluation by replacing complex expressions with polynomial forms.
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Taylor Series

Exponential Function and Parameters

The exponential function e^(ax) depends on the parameter 'a' and variable 'x'. Understanding how the parameter affects the function's behavior and its derivatives is crucial when expressing limits in terms of 'a', especially when using series expansions to evaluate limits involving parameters.
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Exponential Functions
Related Practice
Textbook Question

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = 2x/(1 + x²)²

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Textbook Question

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.


g(x) = − 1/(1 + x)² using f(x) = 1/(1 + x)

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (sin 2x)/x

Textbook Question

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.


f(x) = eᶜᵒˢ ˣ

Textbook Question

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ³⁵ tan ⁻¹x dx

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₀∞ (-x/10)²ᵏ