Explanation Given information: lim ( x , y ) → ( 0 , 0 ) x 2 + y 2 − 2 | x − 1 | + | y − 1 | Calculation: The given f ( x , y ) = x 2 + y 2 − 2 | x − 1 | + | y − 1 | . Since ( x , y ) → ( 1 , 1 ) , so x → 1 ⇒ x − 1 → 0 And y → 1 ⇒ y − 1 → 0 . So let x − 1 = u and y − 1 = v Therefore x = u + 1 and y = v + 1 So the function can be written as lim ( u , v ) → ( 0 , 0 ) ( u + 1 ) 2 + ( v + 1 ) 2 − 2 | u | + | v | As ( u , v ) → ( 0 , 0 ) the limit of the function is calculated as follows, lim ( x , y ) → ( 1 , 1 ) f ( x , y ) = lim ( u , v ) → ( 0 , 0 ) ( u + 1 ) 2 + ( v + 1 ) 2 − 2 | u | + | v | = lim ( u , v ) → ( 0 , 0 ) u 2 + 2 u + 1 + v 2 + 2 v + 1 − 2 | u | + | v | = lim ( u , v ) → ( 0 , 0 ) u 2 + 2 u + v 2 + 2 v | u | + | v | = 0 0 Now we will calculate the limit along the lines u = 0 and v = 0 . Along the line v = 0 lim ( u , v ) → ( 0 , 0 ) u 2 + 2 u + v 2 + 2 v | u | + | v | = lim u → 0 u 2 + 2 u | u | So when u → 0 + , lim u → 0 + u 2 + 2 u | u | = lim u → 0 + u 2 + 2 u u ... | u | = u = lim u → 0 + u ( u + 2 ) u = lim u → 0 + ( u + 2 ) 1 a s u → 0 , u ≠ 0 = 0 + 2 = 2 So when u → 0 − , lim u → 0 − u 2 + 2 u | u | = lim u → 0 − u 2 + 2 u − u

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0…

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Chapter 14.2, Problem 42E

To determine

Toevaluate:

the limit or determine that it does not exist.

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Chapter 14.2, Problem 41E

Chapter 14.2, Problem 43E

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Chapter 14 Solutions

Calculus: Early Transcendentals

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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Toevaluate: the limit or determine that it does not exist.

Solution Summary: The author explains that the limit of the functions underset(x,y)to

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Toevaluate: the limit or determine that it does not exist.

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Chapter 14 Solutions

Toevaluate:

the limit or determine that it does not exist.