   Chapter 14.2, Problem 18E

Chapter
Section
Textbook Problem

Find the limit if it exists, or show that the limit does not exist.18. lim ( x , y ) → ( 0 , 0 ) x y 4 x 2 + y 8

To determine

To find: The limit of the function lim(x,y)(0,0)xy4x2+y8 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)xy4x2+y8 .

Let the path C1 be y=0 and the path C2 be x=y4 .

Evaluate the limit (L1) along the x - axis, that is, y=0 .

lim(x,y)(x,0)xy4x2+y8=lim(x,y)(x,0)x(0)4x2+(0)8=0

Thus, the limit along the curve y=0 is, L1=0

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