   Chapter 14.2, Problem 40E

Chapter
Section
Textbook Problem

Use polar coordinates to find the limit, [If (r, θ) are polar coordinates of the point (x, y) with r ≥ 0, note that r → 0+ as (x, y) → (0, 0).]40. lim ( x , y ) → ( 0 , 0 ) ( x 2 + y 2 ) ln ( x 2 + y 2 )

To determine

To find: The limit of the function lim(x,y)(0,0)(x2+y2)ln(x2+y2) using polar coordinates.

Explanation

Definition used:

“A function f(x,y) is continuous at (a,b) if lim(x,y)(a,b)f(x,y)=f(a,b) ”.

Calculation:

The function is, lim(x,y)(0,0)(x2+y2)ln(x2+y2) .

The polar coordinate of the point (x,y) is (r,θ) where r0 .

Note that r0+ as (x,y)(0,0) .

The value of x and y can be represented in polar coordinates as, x=rcosθ and y=rsinθ

Since every polynomial function is continuous on 2 , the given polynomial function is continuous.

So, substitute x=rcosθ and y=rsinθ directly in the given continuous function and obtain the required limit as shown below.

lim(x,y)(0,0)(x2+y2)ln(x2+y2)=limr0+(r2cos2θ+r2sin2θ)ln(r2cos2θ+r2sin2θ)=<

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