Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. lim xsex x → 00
Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. lim xsex x → 00
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem:**
Find the limit. Use l'Hopital's Rule where appropriate. If there is a more elementary method, consider using it.
\[ \lim_{x \to \infty} x^8 e^{-x^7} \]
**Explanation:**
This problem involves finding the limit of the function \(x^8 e^{-x^7}\) as \(x\) approaches infinity.
- \(x^8\) is a polynomial function of degree 8, which grows without bound as \(x\) increases.
- \(e^{-x^7}\) is an exponential decay function, which approaches zero as \(x\) increases.
The task is to analyze the behavior of this function as \(x\) approaches infinity and determine the limit. You may use l'Hospital's Rule if the limit initially results in an indeterminate form such as \(\frac{\infty}{\infty}\) or \(\frac{0}{0}\). However, consider simpler algebraic techniques for resolving this limit before employing l'Hopital's Rule, particularly given the exponential dominates polynomial growth at infinity in many such cases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0c7c0e9-1042-4b47-9e78-c6aaeaae6479%2F0bee7e44-66a3-4c45-8a57-2108ad7fb0d3%2Ftyf0ve_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the limit. Use l'Hopital's Rule where appropriate. If there is a more elementary method, consider using it.
\[ \lim_{x \to \infty} x^8 e^{-x^7} \]
**Explanation:**
This problem involves finding the limit of the function \(x^8 e^{-x^7}\) as \(x\) approaches infinity.
- \(x^8\) is a polynomial function of degree 8, which grows without bound as \(x\) increases.
- \(e^{-x^7}\) is an exponential decay function, which approaches zero as \(x\) increases.
The task is to analyze the behavior of this function as \(x\) approaches infinity and determine the limit. You may use l'Hospital's Rule if the limit initially results in an indeterminate form such as \(\frac{\infty}{\infty}\) or \(\frac{0}{0}\). However, consider simpler algebraic techniques for resolving this limit before employing l'Hopital's Rule, particularly given the exponential dominates polynomial growth at infinity in many such cases.
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