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f(x, y) approaches the same limiting value 12 as (x,y) → (0,0) along both the x-axis and the y-axis. Which of the following must be true? (A) lim f(x, y) exists and is equal to 12. (x,y) →(0,0)* (C) limf(x,y) exists but we don't know what it is. (x,y) (0,0) lim f(x,y) does not exist. (x,y) →(0,0)* (D) f is not continuous at (0,0) (G) None of the above. (B) (E) f(x, y) will also approach 12 as (x,y) →(0,0) along the path y = x. (F) f(x, y) will not approach 12 as (x, y)→ (0,0) along the path y=x.

f(x, y) approaches the same limiting value 12 as (x,y) → (0,0) along both the x-axis and the y-axis. Which of the following must be true? (A) lim f(x, y) exists and is equal to 12. (x,y) →(0,0)* (C) limf(x,y) exists but we don't know what it is. (x,y) (0,0) lim f(x,y) does not exist. (x,y) →(0,0)* (D) f is not continuous at (0,0) (G) None of the above. (B) (E) f(x, y) will also approach 12 as (x,y) →(0,0) along the path y = x. (F) f(x, y) will not approach 12 as (x, y)→ (0,0) along the path y=x.

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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The problem states: \( f(x, y) \) approaches the same limiting value 12 as \( (x, y) \to (0, 0) \) along both the x-axis and the y-axis. Which of the following must be true? Options: (A) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists and is equal to 12. (B) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) does not exist. (C) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists but we don’t know what it is. (D) \( f \) is not continuous at \( (0, 0) \). (E) \( f(x, y) \) will also approach 12 as \( (x, y) \to (0, 0) \) along the path \( y = x \). (F) \( f(x, y) \) will not approach 12 as \( (x, y) \to (0, 0) \) along the path \( y = x \). (G) None of the above.
Transcribed Image Text:The problem states: \( f(x, y) \) approaches the same limiting value 12 as \( (x, y) \to (0, 0) \) along both the x-axis and the y-axis. Which of the following must be true? Options: (A) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists and is equal to 12. (B) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) does not exist. (C) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists but we don’t know what it is. (D) \( f \) is not continuous at \( (0, 0) \). (E) \( f(x, y) \) will also approach 12 as \( (x, y) \to (0, 0) \) along the path \( y = x \). (F) \( f(x, y) \) will not approach 12 as \( (x, y) \to (0, 0) \) along the path \( y = x \). (G) None of the above.
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