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6. Use a series to find the limit. x-In(1+x) x² a. lim x-0

6. Use a series to find the limit. x-In(1+x) x² a. lim x-0

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Use a Series to Find the Limit ### Problem 6 **a.** \(\lim_{{x \to 0}} \frac{x \ln(1+x)}{x^2}\) **b.** \(\lim_{{x \to 0}} \frac{1-\cos x}{1+x-e^x}\) **Instructions:** For each limit, use the appropriate series expansion to evaluate the limit as \(x\) approaches 0. Consider using Taylor series expansions for functions like \(\ln(1+x)\), \(\cos x\), and \(e^x\) to simplify the expressions first. Apply the limits after the expansions and simplifications to find the final value.
Transcribed Image Text:## Use a Series to Find the Limit ### Problem 6 **a.** \(\lim_{{x \to 0}} \frac{x \ln(1+x)}{x^2}\) **b.** \(\lim_{{x \to 0}} \frac{1-\cos x}{1+x-e^x}\) **Instructions:** For each limit, use the appropriate series expansion to evaluate the limit as \(x\) approaches 0. Consider using Taylor series expansions for functions like \(\ln(1+x)\), \(\cos x\), and \(e^x\) to simplify the expressions first. Apply the limits after the expansions and simplifications to find the final value.
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