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→Limits of Multivariate Functions
The limit of a multivariate function <math>\lim_{(x,y)\rightarrow(x_0,y_0)} f(x,y) \ </math> is almost identical to the single-variable limit in concept, but complicated by the realization that ''(x,y)'' can approach the target point from any direction, not just the left or right hand sides.
The limit of a multivariate function <math>\lim_{(x,y)\rightarrow(x_0,y_0)} f(x,y) \ </math> is almost identical to the single-variable limit in concept, but complicated by the realization that ''(x,y)'' can approach the target point from any direction, not just the left or right hand sides.
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The definition is not dissimilar from the single-variable case. The limit of a function is defined so that <math>\lim_{(x,y) \rightarrow (x_0,y_0)} f(x,y) = L \ </math> if, for any tiny little number <math>\epsilon>0 \ </math> you can think of, there is some distance <math>\delta \ </math> such that every point within <math>\delta \ </math> of <math>(x_0,y_0) \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>.
Nevertheless, the kinds of phenomenon we see in the multivariate case is very different. Note that while it is still possible to define a limit from a direction, we don't bother to do so - unlike the one-variable case when a one-sided deriative told us half of the functions behavior at a point, in higher dimensions such a limit tells us almost nothing. Indeed, the only kind of phenomenon that these kinds of limits are capable of describing are "point discontinuties," which can be "fixed" just be defining the function to be the limit at that point.
=Partial Derivatives=
=Partial Derivatives=