x²y x² + y² does not have a limit at (0, 0) by examining the following limits. (a) Find the limit of ƒ as (x, y) → (0, 0) along the line y = x. lim f(x,y) (z,y) (0,0) = (z,y) (0,0) f(x, y) (b) Find the limit of fas (x, y) → (0,0) along the line y = x². lim f(x,y) = = (Be sure that you are able to explain why the results in (a) and (b) indicate that f does not have a limit at (0,0)!
x²y x² + y² does not have a limit at (0, 0) by examining the following limits. (a) Find the limit of ƒ as (x, y) → (0, 0) along the line y = x. lim f(x,y) (z,y) (0,0) = (z,y) (0,0) f(x, y) (b) Find the limit of fas (x, y) → (0,0) along the line y = x². lim f(x,y) = = (Be sure that you are able to explain why the results in (a) and (b) indicate that f does not have a limit at (0,0)!
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
![The given mathematical problem explores the function
\[ f(x, y) = \frac{x^2 y}{x^4 + y^2}. \]
You need to show that this function does not have a limit at the point (0, 0) by examining the following limits:
(a) Find the limit of \( f \) as \( (x, y) \to (0, 0) \) along the line \( y = x \).
\[
\lim_{(x,y) \to (0,0) \, y=x} f(x, y) = \quad \boxed{}
\]
(b) Find the limit of \( f \) as \( (x, y) \to (0, 0) \) along the line \( y = x^2 \).
\[
\lim_{(x,y) \to (0,0) \, y=x^2} f(x, y) = \quad \boxed{}
\]
**Note:**
(Be sure that you are able to explain why the results in (a) and (b) indicate that \( f \) does not have a limit at (0,0)!)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51597b7e-0432-444a-9bd1-96c29327285f%2Fe05994e2-8bb3-4344-97ba-cb3003e6b32c%2F4scj1wc_processed.png&w=3840&q=75)
Transcribed Image Text:The given mathematical problem explores the function
\[ f(x, y) = \frac{x^2 y}{x^4 + y^2}. \]
You need to show that this function does not have a limit at the point (0, 0) by examining the following limits:
(a) Find the limit of \( f \) as \( (x, y) \to (0, 0) \) along the line \( y = x \).
\[
\lim_{(x,y) \to (0,0) \, y=x} f(x, y) = \quad \boxed{}
\]
(b) Find the limit of \( f \) as \( (x, y) \to (0, 0) \) along the line \( y = x^2 \).
\[
\lim_{(x,y) \to (0,0) \, y=x^2} f(x, y) = \quad \boxed{}
\]
**Note:**
(Be sure that you are able to explain why the results in (a) and (b) indicate that \( f \) does not have a limit at (0,0)!)
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