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1 January, 21:07

Use spherical coordinates to find the limit. [Hint: Let x = rho sin (ϕ) cos (θ), y = rho sin (ϕ) sin (θ), and z = rho cos (ϕ), and note that (x, y, z) → (0, 0, 0) implies rho → 0+.] lim (x, y, z) → (0, 0, 0) arctan 19 x2 + y2 + z2

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  1. 1 January, 22:26
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    lim (x, y, z) → (0, 0, 0) [x*y*z] / (x^2+y^2+z^2) = 0

    Step-by-step explanation:

    First, we need to put the spherical coordinates equations that we will use in our problem

    x = p*sin∅cos Ф

    y = p*sin∅sinФ

    z=p*cos∅

    Then we will state the problem

    lim (x, y, z) → (0, 0, 0) [x*y*z] / (x^2+y^2+z^2)

    Using the spherical coordinates we get

    (x^2+y^2+z^2) = p^2

    Which will make our limit be

    lim p→0 + [p*sin∅cos Ф*p*sin∅sinФ * p*cos∅] / (x^2+y^2+z^2)

    after solving limit:

    lim (x, y, z) → (0, 0, 0) [x*y*z] / (x^2+y^2+z^2) = 0
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