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.B.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.=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):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 f(x,y) must approach L as (x,y) approaches (a,b) along all possible paths. The function f(x,y):OC. 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, thenThe 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 lim f(x,y) must equal f(a,b) if f is continuous. The function f(x,y):(x,y) →(a,b)OC. 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)x + xyxy + 3OC. The statement is false becausexyx+y=2 x = 0 and y=01 x #0 or y#0xy2is a counterexample.is a counterexample.is a counterexample.lim f(x,y) = L.(x,y) →(a,b)A. 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.B. The statement is true because lim f(x,y) = f(a,b).(x,y) →(a,b)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)

Continuity is a fundamental concept in mathematics, particularly in the field of real analysis. It…

Determine whether the following statements are true and give an explanation or 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) O A. 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)= O 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)= OC. The statement is true because if f(x,y) approaches L as (x,y) approaches (a,b) along two different paths in the domain off, then b. If lim f(x,y)=L, then f is continuous at (a,b). Choose the correct interpretation below. (x,y) →(a,b) O A. 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 OB. 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) OC. 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) xy x+y 2 x = 0 and y=0 1 x#0 or y#0 is a counterexample. is a counterexample. is a counterexample. lim f(x,y)=L. (x,y)→(a,b) is a counterexample. OA. 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. OB. The statement is true because lim f(x,y)=f(a,b). (x,y) →→→(a,b) Oc The statement is false because lim fly v) must equal flah) and continuity implies that lim flyv) #fla h). The function f(x y) =tan (x + y) is a counterexample

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.1: Limits

Problem 6YT