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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 11.4.16
Chapter 11, Problem 11.4.16

Limits Evaluate the following limits using Taylor series.
lim ₓ→₄ (x² 16)/(ln (x 3)}

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1
First, rewrite the limit expression clearly: \(\lim_{x \to 4} \frac{x^2 - 16}{\ln(x - 3)}\).
Recognize that as \(x\) approaches 4, the numerator \(x^2 - 16\) approaches \(4^2 - 16 = 0\), and the denominator \(\ln(x - 3)\) approaches \(\ln(1) = 0\), so this is an indeterminate form \(\frac{0}{0}\) suitable for applying Taylor series expansions.
Expand the numerator \(x^2 - 16\) around \(x = 4\) using the Taylor series (or simply use the linear approximation): \(x^2 - 16 = (4)^2 - 16 + 2 \cdot 4 (x - 4) + \cdots = 0 + 8(x - 4) + \cdots\).
Expand the denominator \(\ln(x - 3)\) around \(x = 4\). Since \(x - 3\) approaches 1, use the expansion of \(\ln(1 + h)\) where \(h = x - 4\): \(\ln(x - 3) = \ln(1 + (x - 4)) = (x - 4) - \frac{(x - 4)^2}{2} + \cdots\).
Substitute these expansions back into the limit expression and simplify by canceling common factors, then evaluate the limit by taking \(x \to 4\) (or equivalently \(h \to 0\)).

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Key Concepts

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

Limits and Limit Evaluation

Limits describe the behavior of a function as the input approaches a particular value. Evaluating limits helps determine the function's value near points where direct substitution may be undefined or indeterminate, such as 0/0 or ∞/∞ forms.
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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. It approximates functions near that point, allowing simplification of complex expressions to evaluate limits or analyze behavior.
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Handling Indeterminate Forms Using Series

When direct substitution in limits results in indeterminate forms like 0/0, expanding numerator and denominator into Taylor series helps identify leading terms. This approach simplifies the limit evaluation by canceling common factors and revealing the limit's value.
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Related Practice
Textbook Question

L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.

a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.

b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.

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Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)

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Textbook Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = sin x, a = π/2

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Textbook Question

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

∑ₖ₌₀∞ k(x−1)ᵏ

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Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

√e

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Textbook Question

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

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