Arc Length Problems
Partial Derivatives

Evaluate each limit along the x-axis, y-axis, the line y=x, the curve y=x^2, and the curve y=\sqrt{x}. Explain your result.

(1)   \begin{equation*} \lim _{(x,y) \rightarrow (0,0)} \frac{x^2}{x^2+y^2} \end{equation*}

(2)   \begin{equation*} \lim _{(x,y) \rightarrow (0,0)} \frac{\sqrt{2}x^2y^2}{x^4+y^4} \end{equation*}

(3)   \begin{equation*} \lim _{(x,y) \rightarrow (0,0)} \frac{(x+y)^2}{x^2+y^2} \end{equation*}

(4)   \begin{equation*} \lim _{(x,y) \rightarrow (0,0)} \frac{xy}{\sqrt{x^2+y^2}} \end{equation*}

(5)   \begin{equation*} \lim _{(x,y) \rightarrow (0,0)} \frac{x^2y}{x^4+y^2} \end{equation*}

(6)   \begin{equation*} \lim _{(x,y) \rightarrow (0,0)} \frac{y^3+x^2y}{x^2+y^2} \end{equation*}

To see the notes, visit limit notes.

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