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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![**Evaluate:**
\[
\lim_{{x \to 0}} \frac{{\cos(x) - 1 + \frac{x^2}{2}}}{{12x^4}}
\]
**Hint:** Using power series.
**Explanation:**
This problem requires finding the limit of a function as \( x \) approaches 0. The expression inside the limit involves \(\cos(x)\) and a polynomial term \(\frac{x^2}{2}\), all over \(12x^4\). The hint suggests using power series expansions to simplify the expression before evaluating the limit. The power series for \(\cos(x)\) is:
\[
\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots
\]
Using this series can help simplify the numerator by canceling terms, allowing for easier computation of the limit. The goal is to make the dominant terms in the numerator match the degree of the denominator to evaluate the limit effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2F239008b9-4604-424b-8038-642bf432217a%2F59512m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate:**
\[
\lim_{{x \to 0}} \frac{{\cos(x) - 1 + \frac{x^2}{2}}}{{12x^4}}
\]
**Hint:** Using power series.
**Explanation:**
This problem requires finding the limit of a function as \( x \) approaches 0. The expression inside the limit involves \(\cos(x)\) and a polynomial term \(\frac{x^2}{2}\), all over \(12x^4\). The hint suggests using power series expansions to simplify the expression before evaluating the limit. The power series for \(\cos(x)\) is:
\[
\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots
\]
Using this series can help simplify the numerator by canceling terms, allowing for easier computation of the limit. The goal is to make the dominant terms in the numerator match the degree of the denominator to evaluate the limit effectively.
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