University:
Whitman College
Course:
MATH-225 | Calculus III
Academic year:

2021
Views:

268

Pages:

22
Author:

Angelo Rhodes

Does the limit exist at the origin? f (x, y ) = e−xy cos(x + y ) Does the limit exist at the origin? f (x, y ) = e−xy cos(x + y ) SOLUTION: f is continuous at the origin. The limit is f (0, 0) = 1. Does the limit exist at (1, 0)? 1 + y2 f (x, y ) = ln x 2 + xy Does the limit exist at (1, 0)? 1 + y2 f (x, y ) = ln x 2 + xy SOLUTION: f is continuous at (1, 0). The limit is f (1, 0) = ln(1) = 0. Does the limit exist at (0, 0)? f (x, y ) = x 2 + sin2 (y ) 2x 2 + y 2 Does the limit exist at (0, 0)? f (x, y ) = x 2 + sin2 (y ) 2x 2 + y 2 SOLUTION: Limit along x = 0: lim (x,y )→(0,0) sin(y ) y 2 = Does the limit exist at (0, 0)? f (x, y ) = x 2 + sin2 (y ) 2x 2 + y 2 SOLUTION: Limit along x = 0: lim (x,y )→(0,0) sin(y ) y 2 =1 Does the limit exist at (0, 0)? f (x, y ) = x 2 + sin2 (y ) 2x 2 + y 2 SOLUTION: Limit along x = 0: lim (x,y )→(0,0) sin(y ) y 2 Limit along y = 0: 2 x lim = (x,y )→(0,0) 2x 2 =1 Does the limit exist at (0, 0)? f (x, y ) = x 2 + sin2 (y ) 2x 2 + y 2 SOLUTION: Limit along x = 0: lim (x,y )→(0,0) sin(y ) y 2 =1 Limit along y = 0: 2 1 x lim = 2 2 (x,y )→(0,0) 2x The limit DNE at the origin. Does the limit exist at (0, 0)? f (x, y ) = 6x 3 y 2x 4 + y 4 Does the limit exist at (0, 0)? f (x, y ) = 6x 3 y 2x 4 + y 4 SOLUTION: Limit along y = x: 6x 4 = lim (x,y )→(0,0) 2x 4 + x 4 Does the limit exist at (0, 0)? f (x, y ) = 6x 3 y 2x 4 + y 4 SOLUTION: Limit along y = x: 6x 4 =2 lim (x,y )→(0,0) 2x 4 + x 4 Does the limit exist at (0, 0)? f (x, y ) = 6x 3 y 2x 4 + y 4 SOLUTION: Limit along y = x: 6x 4 =2 lim (x,y )→(0,0) 2x 4 + x 4 Limit along y = −x: −6x 4 lim = (x,y )→(0,0) 2x 4 + x 4 Does the limit exist at (0, 0)? f (x, y ) = 6x 3 y 2x 4 + y 4 SOLUTION: Limit along y = x: 6x 4 =2 lim (x,y )→(0,0) 2x 4 + x 4 Limit along y = −x: −6x 4 lim = −2 (x,y )→(0,0) 2x 4 + x 4 The limit DNE at the origin. Does the limit exist at the origin? f (x, y ) = x4 − y4 x2 + y2 Does the limit exist at the origin? f (x, y ) = x4 − y4 x2 + y2 SOLUTION: The numerator can be factored Does the limit exist at the origin? f (x, y ) = x4 − y4 x2 + y2 SOLUTION: The numerator can be factored (x 2 + y 2 )(x 2 − y 2 ) = x2 − y2 x2 + y2 Does the limit exist at the origin? f (x, y ) = x4 − y4 x2 + y2 SOLUTION: The numerator can be factored (x 2 + y 2 )(x 2 − y 2 ) = x2 − y2 x2 + y2 so the limit is zero. Does the limit exist at the origin? f (x, y ) = x 2 sin2 (y ) x 2 + 2y 2 Does the limit exist at the origin? f (x, y ) = SOLUTION: x 2 sin2 (y ) x 2 + 2y 2 Does the limit exist at the origin? f (x, y ) = x 2 sin2 (y ) x 2 + 2y 2 SOLUTION: Squeeze it! 0≤ x2 x 2 sin2 (y ) ≤ 2 ≤ x2 2 2 x + 2y x + 2y 2 By the Squeeze Theorem, the limit is zero.

Calculus III Solutions Section 14.4 Story Problems

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