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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Question
12) find the limit
![The image contains a mathematical expression written on lined paper. The expression is:
\[
\lim_{x \to 0} \left(1 + \frac{4}{x^3}\right)^x
\]
### Explanation:
This expression represents a limit in calculus as \( x \) approaches 0. Specifically, it examines the behavior of the function \(\left(1 + \frac{4}{x^3}\right)^x\) as \(x\) gets closer and closer to zero. Such limits are fundamental in the study of calculus, particularly in understanding behaviors that can't be easily calculated by direct substitution. The presence of an exponent of \( x \) and a term \(\frac{4}{x^3}\) suggests the need for careful analysis, potentially using L'Hôpital's Rule or logarithmic properties to evaluate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc92c6411-e0c2-4822-8067-9688589da753%2Fe6100194-150b-461b-a057-35fe97f38dc4%2F01tfas_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression written on lined paper. The expression is:
\[
\lim_{x \to 0} \left(1 + \frac{4}{x^3}\right)^x
\]
### Explanation:
This expression represents a limit in calculus as \( x \) approaches 0. Specifically, it examines the behavior of the function \(\left(1 + \frac{4}{x^3}\right)^x\) as \(x\) gets closer and closer to zero. Such limits are fundamental in the study of calculus, particularly in understanding behaviors that can't be easily calculated by direct substitution. The presence of an exponent of \( x \) and a term \(\frac{4}{x^3}\) suggests the need for careful analysis, potentially using L'Hôpital's Rule or logarithmic properties to evaluate.
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