Explanation Given information: a , b ≥ 0 lim ( x , y ) → ( 0 , 0 ) x a y b x 2 + y 2 Calculation: We will find the limit along x -axis. We know that y = 0 on the x -axis. So lim x → 0 f ( x , 0 ) = lim x → 0 x a ( 0 ) b x 2 + ( 0 ) 2 = 0 Hence as ( x , y ) → ( 0 , 0 ) , f ( x , y ) → 0 along x -axis. Now we will calculate limit along y -axis. So lim y → 0 f ( 0 , y ) = lim y → 0 ( 0 ) a ( y ) b 0 2 + ( y ) 2 = 0 Hence as ( x , y ) → ( 0 , 0 ) , f ( x , y ) → 0 along y -axis

Loose-leaf Version For Calculus: Early Transcendentals

4th Edition

ISBN: 9781319055912

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…

Question

Chapter 14.2, Problem 44E

To determine

Toshow:

that lim(x,y)→(0,0)xaybx2+y2=0 if a+b>2 and that the limit does not exist if a+b≤2.

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Chapter 14.2, Problem 43E

Chapter 14.2, Problem 45E

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Chapter 14 Solutions

Loose-leaf Version For 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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Toshow: that lim ( x , y ) → ( 0 , 0 ) x a y b x 2 + y 2 = 0 if a + b > 2 and that the limit does not exist if a + b ≤ 2 .

Solution Summary: The author explains that underset(x,y)to 0mathrmlimxay

Concept explainers

Toshow:

that lim(x,y)→(0,0)xaybx2+y2=0 if a+b>2 and that the limit does not exist if a+b≤2.

Want to see the full answer?

Chapter 14 Solutions

Toshow: that lim ( x , y ) → ( 0 , 0 ) x a y b x 2 + y 2 = 0 if a + b &gt; 2 and that the limit does not exist if a + b ≤ 2 .

Solution Summary: The author explains that underset(x,y)to 0mathrmlimxay

Concept explainers

Toshow: that lim(x,y)→(0,0)xaybx2+y2=0 if a+b>2 and that the limit does not exist if a+b≤2.

Want to see the full answer?

Chapter 14 Solutions

Toshow: that lim ( x , y ) → ( 0 , 0 ) x a y b x 2 + y 2 = 0 if a + b > 2 and that the limit does not exist if a + b ≤ 2 .

Toshow:

that lim(x,y)→(0,0)xaybx2+y2=0 if a+b>2 and that the limit does not exist if a+b≤2.