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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Evaluate the limit or show that it does not exist.

10. lim ( x , y ) ( 0 , 0 ) 2 x y x 2 + 2 y 2

To determine

To find: The limit of the function lim(x,y)(0,0)(2xyx2+2y2) if exist, otherwise show that the limit does not exist.

Explanation

Result used:

“If f(x,y)L1 as (x,y)(a,b) along a path C1 and f(x,y)L2 as (x,y)(a,b) along a path C2 , where L1L2 , then lim(x,y)(a,b)f(x,y) does not exist”.

Calculation:

Consider the given function, lim(x,y)(0,0)(2xyx2+2y2) .

Let the path C1 be y=x and the path C2 be y=x2 .

Evaluate the limit (L1) along the curve y=x .

lim(x,y)(0,0)(2xyx2+2y2)=lim(x,y)(0,0)2x(x)x2+2(x)2=lim(x,y)(0,0)2x2x2+2x2=lim(x,y)(0,0)2x23x2=23

Thus, the limit along the curve y=x is L1=23

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Chapter 14 Solutions

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