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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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