Find the limit of f as (x , y) → (0 ,0 )or show that the limit does not exist. Consider converting the function to polar coordinates to make

Question

Find the limit of f as (x , y) → (0 ,0 )or show that the limit does not exist. Consider converting the function to polar coordinates to make finding the limit easier.
f(x , y) = (x³ – xy²)/ (x² + y²)

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Jade 1 week 2021-09-15T14:42:11+00:00 1 Answer 0

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    2021-09-15T14:43:36+00:00

    As suggested, let’s use polar coordinates:

    \begin{cases}x=\rho\cos(\theta)\\y=\rho\sin(\theta)\end{cases}

    To get

    f(x,y)=\dfrac{x^3-xy^2}{x^2+y^2}\mapsto \dfrac{\rho^3\cos^3(\theta)-\rho^3\cos(\theta)\sin^2(\theta)}{\rho^2}

    We can simplify the expression to

    \dfrac{\rho^3\cos^3(\theta)-\rho^3\cos(\theta)\sin^2(\theta)}{\rho^2}=\dfrac{\rho^3\cos(\theta)(\cos^2(\theta)-\sin^2(\theta))}{\rho^2}

    And simplify \rho^2 to get

    \rho\cos(\theta)(\cos^2(\theta)-\sin^2(\theta))

    So, as \rho\to 0, f(x,y)\to 0 as well.

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