Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim (x, y)→(0, 0) xy/square root x2 + y2

College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter3: Polynomial And Rational Functions
Section3.6: Rational Functions
Problem 2E
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Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim (x, y)→(0, 0) xy/square root x2 + y2
Expert Solution
Step 1

The given limit is lim(x,y)(0,0)xyx2+y2.

Substitute x=rcosθ,y=rsinθ in the above equation.

lim(x,y)(0,0)xyx2+y2=limr0rcosθrsinθr2cos2θ+r2sin2θlim(x,y)(0,0)xyx2+y2=limr0r2cosθsinθr2cos2θ+sin2θlim(x,y)(0,0)xyx2+y2=limr0r2cosθsinθr2lim(x,y)(0,0)xyx2+y2=cosθsinθ limr0rlim(x,y)(0,0)xyx2+y2=0

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