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Here is an example with your function. Fix $\alpha=1$ and imagine in rectangular coordinates, approaching the limit along the $x$-axis (so $y=0$). Then you get $$\frac{x^2}{x^2+y^2} \to \frac{x^2}{x^2+0} = 1.$$ Now imagine approaching from the $y$-axis, so $x=0$. You get $$\frac{x^2}{x^2+y^2} \to \frac{0}{0+y^2} = 0.$$ Since the values disagree, the ...

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There's no upper bound for the average speed. The speed is determined by the height; the deeper you go, the higher you can make the average speed.

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Consider the two points at which the tangent planes touch the paraboloid: $\left(x_1, y_1,\frac{x_1^2+y_1^2}{2}\right)$ and $\left(x_2, y_2,\frac{x_2^2+y_2^2}{2}\right)$ The tangent planes to these points are: $z-\frac{x_1^2+y_1^2}{2}=x_1(x-x_1)+y_1(y-y_1)$ and $z-\frac{x_2^2+y_2^2}{2}=x_2(x-x_2)+y_2(y-y_2)$. (This is from the equation $z-z_0=f_x(x_0,... 1 Let the tangent planes touch the surface at$(x', y',z')$and$(x'',y'',z'')$, $$\left \{ \begin{array}{rcl} x'x+y'y &=& z+z' \\ x''x+y''y &=& z+z'' \end{array} \right.$$ The line of intersection$(\xi, \eta, \zeta)\$ is $$(\xi, \eta, \zeta)= \left( \frac{\begin{vmatrix} \zeta+z' & y' \\ \zeta+z'' & y'' \end{vmatrix}} ... 1 The answer to your question hinges on what exactly \alpha is supposed to be. If \alpha is a parameter which defines a family of functions, there may be some values of \alpha for which the function is continuous everywhere. It certainly is not continuous for all \alpha in either \mathbb{R} or \mathbb{Z}, since \alpha = 0 produces the function$$ ...

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