Question

What is the limit of (x^3+y^3)/(x^2+y^2)   as (x,y)-->(0,0)? I'm supposed to use polar coordinates to find the limit. 
The assignment says: "If (r,theta ) are polar coordinates of the point (x,y) with r > 0 (or equal to 0), note that r --> 0 as (x,y) --> (0,0)."

Accepted Answer

The problem requests to use parametric forms, hence, you need to convert the Cartesian coordinates x,y into polar coordinates, thus, you need to substitute r*cos t  for x and r*sin t  for y, such that:
lim_(x,y -> 0) (x^3 + y^3)/(x^2 + y^2) = lim_(r->0) (r^3*cos^3 t + r^3*sin^3 t)/(r^2*cos^2 t + r^2*sin^2 t)
Factoring out r^3 and r^2  yields:
lim_(r->0) (r^3*(cos^3 t+sin^3 t))/(r^2*(cos^2 t+sin^2 t))
Using the basic formula of trigonometry, cos^2 t+sin^2 t = 1 , yields:
lim_(r->0) r*(cos^3 t+sin^3 t) = 0
Hence, evaluating the given multivariable limit, using parametric forms, yields lim_(x,y -> 0) (x^3 + y^3)/(x^2 + y^2) = 0.

Answer

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What is the limit of `(x^3+y^3)/(x^2+y^2)` as (x,y)-->(0,0)? I'm supposed to use polar coordinates to find the limit. The assignment says: "If (`r,theta` ) are polar coordinates of the point (x,y) with r > 0 (or equal to 0), note that r --> 0 as (x,y) --> (0,0)."

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