Problem 5.3 Limits and continuity a) Find the limits of 06 ( 12² + 1²1 ) 1+x+y lim COS (x,y) (0,0) b) Find the following limits by rewriting the fraction first: x-y lim (x,y) → (2,2) x4-y4' and lim (x,y) (0,0) x4 + y² and and xy lim (x,y) →(5,1) x+y c) By considering different paths of approach, show that the following limits do not exist: xy³ lim (x,y) (0,0) x2 + yº· x+y-4 (x,y)+(2,2) √√x+y=2' lim
Problem 5.3 Limits and continuity a) Find the limits of 06 ( 12² + 1²1 ) 1+x+y lim COS (x,y) (0,0) b) Find the following limits by rewriting the fraction first: x-y lim (x,y) → (2,2) x4-y4' and lim (x,y) (0,0) x4 + y² and and xy lim (x,y) →(5,1) x+y c) By considering different paths of approach, show that the following limits do not exist: xy³ lim (x,y) (0,0) x2 + yº· x+y-4 (x,y)+(2,2) √√x+y=2' lim
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,
Can someone please show how to solve this problem?
Thank you
![**Problem 5.3: Limits and Continuity**
a) Find the limits of
\[
\lim_{(x,y) \to (0,0)} \cos \left( \frac{x^2 + y^3}{1 + x + y} \right)
\]
and
\[
\lim_{(x,y) \to (5,1)} \frac{xy}{x + y}.
\]
b) Find the following limits by rewriting the fraction first:
\[
\lim_{(x,y) \to (2,2)} \frac{x-y}{x^4 - y^4},
\]
and
\[
\lim_{(x,y) \to (2,2)} \frac{x + y - 4}{\sqrt{x + y - 2}}.
\]
c) By considering different paths of approach, show that the following limits do not exist:
\[
\lim_{(x,y) \to (0,0)} \frac{x^4}{x^4 + y^2},
\]
and
\[
\lim_{(x,y) \to (0,0)} \frac{xy^3}{x^2 + y^6}.
\]
d) At what points \((x,y)\) in the plane are the functions \(f(x,y,z)\) continuous?
\[
f(x,y,z) = \frac{1}{|xy| + |z|},
\]
and
\[
f(x,y,z) = \ln(xyz).
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ccc228-61ae-45bc-bff3-36e89ce5abe9%2F80fe34e1-c555-4925-b1d8-6863f47e4d11%2F5wkfq5d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5.3: Limits and Continuity**
a) Find the limits of
\[
\lim_{(x,y) \to (0,0)} \cos \left( \frac{x^2 + y^3}{1 + x + y} \right)
\]
and
\[
\lim_{(x,y) \to (5,1)} \frac{xy}{x + y}.
\]
b) Find the following limits by rewriting the fraction first:
\[
\lim_{(x,y) \to (2,2)} \frac{x-y}{x^4 - y^4},
\]
and
\[
\lim_{(x,y) \to (2,2)} \frac{x + y - 4}{\sqrt{x + y - 2}}.
\]
c) By considering different paths of approach, show that the following limits do not exist:
\[
\lim_{(x,y) \to (0,0)} \frac{x^4}{x^4 + y^2},
\]
and
\[
\lim_{(x,y) \to (0,0)} \frac{xy^3}{x^2 + y^6}.
\]
d) At what points \((x,y)\) in the plane are the functions \(f(x,y,z)\) continuous?
\[
f(x,y,z) = \frac{1}{|xy| + |z|},
\]
and
\[
f(x,y,z) = \ln(xyz).
\]
Expert Solution
Step 1
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 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,
