   Chapter 14, Problem 5RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x, y) ⟶ L as (x, y) ⟶ (a, b) along every straight line through (a, b), then lim ( x , y ) → ( a , b ) f ( x , y ) = L .

To determine

Whether the statement “If f(x,y)L as (x,y)(a,b) along every straight line through (a,b) , then lim(x,y)(a,b)f(x,y)=L ” is true or false.

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:

The given statement is false since the below example disproves the given statement.

Consider the function, lim(x,y)(0,0)(5y4cos2xx4+y4) .

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

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

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