a. If the limits lim f(x,0) and lim f(0,y) exist and equal L, then lim f(x,y) = L. Choose the correct interpretation below. (x,0)→(0,0) (0.y)→ (0,0) (x,y) → (0,0) A. xy The statement is false because f(x,y) must approach L as (x,y) approaches (a,b) along all possible paths. The function f(x,y) = XY OB. The statement is true because if f(x,y) approaches Las (x,y) approaches (a,b) along two different paths in the domain of f, then O C. The statement is false because f(x,y) must approach Las (x,y) approaches (a,b) along all possible paths. The function f(x,y)= b. If lim f(x,y)=L, then f is continuous at (a,b). Choose the correct interpretation below. (x,y) →(a,b) A. B. The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y)= (x,y)→(a,b) 2 x = 0 and y=0 1 x#0 or y#0 The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y) = (x,y) →(a,b) x + xy xy +3 OC. The statement is true because any function f is continuous provided lim f(x,y) exists. (x,y)→(a,b) is a counterexample. lim f(x,y)=L. (x,y)→(a,b) xy x+y is a counterexample. is a counterexample. is a counterexample.
a. If the limits lim f(x,0) and lim f(0,y) exist and equal L, then lim f(x,y) = L. Choose the correct interpretation below. (x,0)→(0,0) (0.y)→ (0,0) (x,y) → (0,0) A. xy The statement is false because f(x,y) must approach L as (x,y) approaches (a,b) along all possible paths. The function f(x,y) = XY OB. The statement is true because if f(x,y) approaches Las (x,y) approaches (a,b) along two different paths in the domain of f, then O C. The statement is false because f(x,y) must approach Las (x,y) approaches (a,b) along all possible paths. The function f(x,y)= b. If lim f(x,y)=L, then f is continuous at (a,b). Choose the correct interpretation below. (x,y) →(a,b) A. B. The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y)= (x,y)→(a,b) 2 x = 0 and y=0 1 x#0 or y#0 The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y) = (x,y) →(a,b) x + xy xy +3 OC. The statement is true because any function f is continuous provided lim f(x,y) exists. (x,y)→(a,b) is a counterexample. lim f(x,y)=L. (x,y)→(a,b) xy x+y is a counterexample. is a counterexample. is a counterexample.
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
![Determine whether the following statements are true and give an explanation or counterexample.
a. If the limits lim f(x,0) and
(x,0)→(0,0)
A.
xy
2
The statement is false because f(x,y) must approach Las (x,y) approaches (a,b) along all possible paths. The function f(x, y) =
B. The statement is true because if f(x,y) approaches L as (x,y) approaches (a,b) along two different paths in the domain of f, then
lim f(0,y) exist and equal L, then lim f(x,y)= L. Choose the correct interpretation below.
(0,y) → (0,0)
(x,y)→ (0,0)
A.
b. If
lim f(x,y) = L, then f is continuous at (a,b). Choose the correct interpretation below.
(x,y) →(a,b)
B.
The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y) =
(x,y) →(a,b)
The statement is false because f(x,y) must approach L as (x,y) approaches (a,b) along all possible paths. The function f(x,y) =
xy
x+y
The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y) =
(x,y) →(a,b)
x + xy
xy + 3
C. The statement is true because any function f is continuous provided lim f(x,y) exists.
(x,y) →(a,b)
c. If f is continuous at (a,b), then lim f(x,y) exists. Choose the correct interpretation below.
(x,y) →(a,b)
2 x = 0 and y = 0
1 x #0 or y#0
A. The statement is false because
is a counterexample.
lim f(x,y) = L.
(x,y) → (a,b)
is a counterexample.
is a counterexample.
is a counterexample.
lim f(x,y) must equal f(a,b) and continuity implies that lim f(x,y) #f(a,b). The function f(x,y)=tan (x + y) is a counterexample.
(x,y) →→(a,b)
(x,y)→ (a,b)
B. The statement is true because lim f(x,y) = f(a,b).
(x,y) →(a,b)
C. The statement is false because continuity only implies that f is defined at (a,b). The function f(x,y) = tan (x + y) is a counterexample.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ade5d6c-ff37-4df2-b1de-522b52697cc9%2F47bbc94f-fb37-45ac-a31e-fe6533fb03a7%2Fpbjafjx_processed.png&w=3840&q=75)
Transcribed Image Text:Determine whether the following statements are true and give an explanation or counterexample.
a. If the limits lim f(x,0) and
(x,0)→(0,0)
A.
xy
2
The statement is false because f(x,y) must approach Las (x,y) approaches (a,b) along all possible paths. The function f(x, y) =
B. The statement is true because if f(x,y) approaches L as (x,y) approaches (a,b) along two different paths in the domain of f, then
lim f(0,y) exist and equal L, then lim f(x,y)= L. Choose the correct interpretation below.
(0,y) → (0,0)
(x,y)→ (0,0)
A.
b. If
lim f(x,y) = L, then f is continuous at (a,b). Choose the correct interpretation below.
(x,y) →(a,b)
B.
The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y) =
(x,y) →(a,b)
The statement is false because f(x,y) must approach L as (x,y) approaches (a,b) along all possible paths. The function f(x,y) =
xy
x+y
The statement is false because lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y) =
(x,y) →(a,b)
x + xy
xy + 3
C. The statement is true because any function f is continuous provided lim f(x,y) exists.
(x,y) →(a,b)
c. If f is continuous at (a,b), then lim f(x,y) exists. Choose the correct interpretation below.
(x,y) →(a,b)
2 x = 0 and y = 0
1 x #0 or y#0
A. The statement is false because
is a counterexample.
lim f(x,y) = L.
(x,y) → (a,b)
is a counterexample.
is a counterexample.
is a counterexample.
lim f(x,y) must equal f(a,b) and continuity implies that lim f(x,y) #f(a,b). The function f(x,y)=tan (x + y) is a counterexample.
(x,y) →→(a,b)
(x,y)→ (a,b)
B. The statement is true because lim f(x,y) = f(a,b).
(x,y) →(a,b)
C. The statement is false because continuity only implies that f is defined at (a,b). The function f(x,y) = tan (x + y) is a counterexample.
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