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Using polar coordinates to determine the limit

Lim (x, y)->(0,0)(X^3+y^3)/(x^2+y^2)

The answer is -1, but I can't get it there. Here is what I did.
((Rcosx)^3 +(rsinx)^3)/((rcosx)^2+(rsinx)^2)
Then by factoring out a r squared from top and bottom I'm left with a denominator of (sin^2(x ) + cos^2 (x)) which simplifies to 1. And a numerator of:
Rcos^3(x)+rsin^3 (x)

This is about as far as I've gotten and I can't figure out how to get it to a -1.
 

Zondrina

Homework Helper
2,063
136
Lim (x, y)->(0,0)(X^3+y^3)/(x^2+y^2)

The answer is -1, but I can't get it there. Here is what I did.
((Rcosx)^3 +(rsinx)^3)/((rcosx)^2+(rsinx)^2)
Then by factoring out a r squared from top and bottom I'm left with a denominator of (sin^2(x ) + cos^2 (x)) which simplifies to 1. And a numerator of:
Rcos^3(x)+rsin^3 (x)

This is about as far as I've gotten and I can't figure out how to get it to a -1.
The answer is certainly not -1. The answer is 0 as ##r → 0##.

Conversely, if ##\epsilon > 0## is given, then choose ##\delta = \frac{\epsilon}{2}##.
 

ehild

Homework Helper
15,360
1,766
The limit is not -1. If it were, the function should get closer and closer to it, when (x,y) gets closer and closer to (0,0). If you choose x=y, f(x,x)=x →0. If you choose x=rcosθ, y=rsinθ, f=r(cos3θ+sin3θ) which goes to 0 if r→0, for any value of θ.

ehild
 

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