lim,** 3 5n-2 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Hello there, can you help me solve problems? Thank you!
Use the definition of Convergent Sequences to prove:
![The image shows a mathematical expression related to limits in calculus:
\[ \lim_{{n \to \infty}} \frac{3}{5n - 2} = 0 \]
This expression states that as \( n \) approaches infinity, the value of the fraction \(\frac{3}{5n - 2}\) approaches 0. This illustrates the concept of limits, where the denominator grows much larger than the numerator, causing the fraction as a whole to diminish towards zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F23470d16-686d-4cc5-a6ec-73a63f691f0c%2Fgu1yo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical expression related to limits in calculus:
\[ \lim_{{n \to \infty}} \frac{3}{5n - 2} = 0 \]
This expression states that as \( n \) approaches infinity, the value of the fraction \(\frac{3}{5n - 2}\) approaches 0. This illustrates the concept of limits, where the denominator grows much larger than the numerator, causing the fraction as a whole to diminish towards zero.
![This image presents a mathematical expression involving limits. The expression reads:
\[
\lim_{{n \to \infty}} \frac{3n + 1}{7n - 3} = \frac{3}{7}
\]
Explanation:
- The expression denotes the limit of a rational function as \( n \) approaches infinity.
- The numerator of the rational function is \( 3n + 1 \) and the denominator is \( 7n - 3 \).
- As \( n \) increases without bound, the key terms that dominate in both the numerator and the denominator are those with \( n \), i.e., \( 3n \) and \( 7n \).
- Dividing both the numerator and the denominator by \( n \) simplifies the expression to \( \frac{3 + \frac{1}{n}}{7 - \frac{3}{n}} \).
- As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{3}{n} \) approach 0.
- Therefore, the expression simplifies to \( \frac{3}{7} \), which is the limiting value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F23470d16-686d-4cc5-a6ec-73a63f691f0c%2Fj8m6jem_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This image presents a mathematical expression involving limits. The expression reads:
\[
\lim_{{n \to \infty}} \frac{3n + 1}{7n - 3} = \frac{3}{7}
\]
Explanation:
- The expression denotes the limit of a rational function as \( n \) approaches infinity.
- The numerator of the rational function is \( 3n + 1 \) and the denominator is \( 7n - 3 \).
- As \( n \) increases without bound, the key terms that dominate in both the numerator and the denominator are those with \( n \), i.e., \( 3n \) and \( 7n \).
- Dividing both the numerator and the denominator by \( n \) simplifies the expression to \( \frac{3 + \frac{1}{n}}{7 - \frac{3}{n}} \).
- As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{3}{n} \) approach 0.
- Therefore, the expression simplifies to \( \frac{3}{7} \), which is the limiting value.
Expert Solution
Step 1: Notes
Since you have posted a multiple question according to guidelines I will solve first question for you. To get remaining part solved please repost the complete question and mention parts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
