Show that the limits do not exist
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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Show that the limits do not exist
![The image shows the following mathematical expression for calculating a limit:
\[
\lim_{(x,y) \to (0,0)} \frac{y + \sin x}{x + \sin y}
\]
This expression represents the limit of the function \(\frac{y + \sin x}{x + \sin y}\) as the point \((x, y)\) approaches the origin \((0, 0)\) in the Cartesian plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5563205c-2551-4780-a535-119e8d9f322e%2Fa1f1d31a-78a4-4e72-8e75-dd2d0c62ba2c%2Facfdoj9_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows the following mathematical expression for calculating a limit:
\[
\lim_{(x,y) \to (0,0)} \frac{y + \sin x}{x + \sin y}
\]
This expression represents the limit of the function \(\frac{y + \sin x}{x + \sin y}\) as the point \((x, y)\) approaches the origin \((0, 0)\) in the Cartesian plane.
![The image shows a mathematical expression representing a limit. The expression is:
\[
\lim\limits_{(x,y) \to (1,0)} \frac{xe^y - 1}{xe^y - 1 + y}
\]
Here, the limit is taken as the point \((x, y)\) approaches \((1, 0)\). The numerator of the fraction is \(xe^y - 1\), and the denominator is \(xe^y - 1 + y\). This type of limit is often evaluated to explore the behavior of a function near a particular point and may involve techniques such as L'Hôpital's rule or algebraic simplification.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5563205c-2551-4780-a535-119e8d9f322e%2Fa1f1d31a-78a4-4e72-8e75-dd2d0c62ba2c%2F5lklhen_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical expression representing a limit. The expression is:
\[
\lim\limits_{(x,y) \to (1,0)} \frac{xe^y - 1}{xe^y - 1 + y}
\]
Here, the limit is taken as the point \((x, y)\) approaches \((1, 0)\). The numerator of the fraction is \(xe^y - 1\), and the denominator is \(xe^y - 1 + y\). This type of limit is often evaluated to explore the behavior of a function near a particular point and may involve techniques such as L'Hôpital's rule or algebraic simplification.
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