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EXAMPLE 3 If f(x, y) = 3x2 + y does lim (x, y) - (0, 0) f(x, y) exist? SOLUTION Let's try to save some time by letting (x, y) → (0, 0) alc nonvertical line through the origin. Then y = mx, where m is the slop 2( (x, y) = f(x, mx) =

EXAMPLE 3 If f(x, y) = 3x2 + y does lim (x, y) - (0, 0) f(x, y) exist? SOLUTION Let's try to save some time by letting (x, y) → (0, 0) alc nonvertical line through the origin. Then y = mx, where m is the slop 2( (x, y) = f(x, mx) =

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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Video Example ) 2xy2 3x2 + yA EXAMPLE 3 If f(x, y) = does lim (x, y)- (0, 0) f(x, y) exist? SOLUTION Let's try to save some time by letting (x, y) → (0, 0) along any nonvertical line through the origin. Then y = mx, where m is the slope, and f(x, у) %3D f(x, тх) %3 3x² + (mx)ª 3x2 + mªx4 3 + m*x So f(x, y) → as (x, y) → (0, 0) along y = mx. Thus f has the same limiting value along every nonvertical line through the origin. But that does not show that the given limit is 0, for if we now let (x, y) → (0, 0) along the parabola x = y?, we have f(x, y) = f(y², y) = 3(y²)² + yA 4y4 So f(x, y) → as (x, y) → (0, 0) along x = y². Since different paths lead to different limiting values, the given limit does not exist.
Transcribed Image Text:Video Example ) 2xy2 3x2 + yA EXAMPLE 3 If f(x, y) = does lim (x, y)- (0, 0) f(x, y) exist? SOLUTION Let's try to save some time by letting (x, y) → (0, 0) along any nonvertical line through the origin. Then y = mx, where m is the slope, and f(x, у) %3D f(x, тх) %3 3x² + (mx)ª 3x2 + mªx4 3 + m*x So f(x, y) → as (x, y) → (0, 0) along y = mx. Thus f has the same limiting value along every nonvertical line through the origin. But that does not show that the given limit is 0, for if we now let (x, y) → (0, 0) along the parabola x = y?, we have f(x, y) = f(y², y) = 3(y²)² + yA 4y4 So f(x, y) → as (x, y) → (0, 0) along x = y². Since different paths lead to different limiting values, the given limit does not exist.
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