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Mathematics LibreTexts

6E: Review Excersies

  • Page ID
    31882
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    Exercise \(\PageIndex{1}\)

    Evaluate the indicated limit or explain why it does not exist.

    1) \(\displaystyle \lim_{(x,y)\to(0,0)} {2\sqrt{x^2+y^2}}\)

    2) \(\displaystyle \lim_{(x,y)\to(0,0)} \frac{3x}{x^2+y^2}\)

    3) \(\displaystyle \lim_{(x,y)\to(0,0)} \frac{2\sin(xy)}{x^2+y^2}\)

    4) \(\displaystyle \lim_{(x,y)\to(0,0)} \frac{4x^2y^2}{x^2+y^4}\)

    Answer

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    Exercise \(\PageIndex{2}\)

    Define \(f(0,0)\) in a way that extends \[f(x,y)=2xy \frac{x^2-y^2}{x^2+y^2}\[ to be continuous at the origin.

    Answer

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    Exercise \(\PageIndex{3}\)

    Find the first partial derivative of \[f(x,y,z)=3x^{(y\ln z)}\[ at \((e,2,e)\).

    Answer

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