   Chapter 14.2, Problem 21E

Chapter
Section
Textbook Problem

Find the limit if it exists, or show that the limit does not exist.21. lim ( x , y , z ) → ( 0 , 0 , 0 ) x y + y z 2 + x z 2 x 2 + y 2 + z 4

To determine

To find: The limit of the function lim(x,y.z)(0,0,0)xy+yz2+xz2x2+y2+z4 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.z)(0,0,0)xy+yz2+xz2x2+y2+z4 .

Let the path C1 be x=0 and y=0 and the path C2 be y=x and z2=x .

Evaluate the limit (L1) along the z - axis, x=0 and y=0 .

lim(x,y,z)(0,0,0)xy+yz2+xz2x2+y2+z4=lim(x,y,z)(0,0,0)(0)(0)+(0)z2+(0)z2(0)<

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