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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

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

Advanced Engineering Mathematics
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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**Determine whether the following statements are true and give an explanation or counterexample:** --- **a. If the limits \( \lim_{(x,0) \to (0,0)} f(x,0) \) and \( \lim_{(0,y) \to (0,0)} f(0,y) \) exist and equal \( L \), then \( \lim_{(x,y) \to (0,0)} f(x,y) = L \). Choose the correct interpretation below.** - **A.** The statement is false because \( f(x,y) \) must approach \( L \) as \((x,y)\) approaches \((a,b)\) along all possible paths. The function \( f(x,y) = \frac{xy}{x+y} \) is a counterexample. - **B.** The statement is false because \( f(x,y) \) must approach \( L \) as \((x,y)\) approaches \((a,b)\) along all possible paths. The function \( f(x,y) = \frac{xy}{2} \) is a counterexample. - **C.** 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 \( \lim_{(x,y) \to (a,b)} f(x,y) = L \). --- **b. If \( \lim_{(x,y) \to (a,b)} f(x,y) = L \), then \( f \) is continuous at \((a,b)\). Choose the correct interpretation below.** - **A.** The statement is false because \( \lim_{(x,y) \to (a,b)} f(x,y) \) must equal \( f(a,b) \) if \( f \) is continuous. The function \( f(x,y) = \frac{x + xy}{xy + 3} \) is a counterexample. - **B.** The statement is false because \( \lim_{(x,y) \to (a,b)} f(x,y) \) must equal \( f(a,b) \) if \( f \) is continuous. The function \( f(x,y) = \begin{cases} 2 &
Transcribed Image Text:**Determine whether the following statements are true and give an explanation or counterexample:** --- **a. If the limits \( \lim_{(x,0) \to (0,0)} f(x,0) \) and \( \lim_{(0,y) \to (0,0)} f(0,y) \) exist and equal \( L \), then \( \lim_{(x,y) \to (0,0)} f(x,y) = L \). Choose the correct interpretation below.** - **A.** The statement is false because \( f(x,y) \) must approach \( L \) as \((x,y)\) approaches \((a,b)\) along all possible paths. The function \( f(x,y) = \frac{xy}{x+y} \) is a counterexample. - **B.** The statement is false because \( f(x,y) \) must approach \( L \) as \((x,y)\) approaches \((a,b)\) along all possible paths. The function \( f(x,y) = \frac{xy}{2} \) is a counterexample. - **C.** 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 \( \lim_{(x,y) \to (a,b)} f(x,y) = L \). --- **b. If \( \lim_{(x,y) \to (a,b)} f(x,y) = L \), then \( f \) is continuous at \((a,b)\). Choose the correct interpretation below.** - **A.** The statement is false because \( \lim_{(x,y) \to (a,b)} f(x,y) \) must equal \( f(a,b) \) if \( f \) is continuous. The function \( f(x,y) = \frac{x + xy}{xy + 3} \) is a counterexample. - **B.** The statement is false because \( \lim_{(x,y) \to (a,b)} f(x,y) \) must equal \( f(a,b) \) if \( f \) is continuous. The function \( f(x,y) = \begin{cases} 2 &
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