Unformatted text preview: it has a limit along some line (or even every line) through the point. The following example shows the folly of this point of view: consider the function de±ned for −→ x n = −→ ∈ R 2 by f ( x,y ) = xy x 2 + y 2 , ( x,y ) n = (0 , 0) If we look at the values of the function along a line through −→ x = −→ 0 of slope m , y = mx, we see that the values of f ( −→ x ) at points on this line are f ( x,mx ) = ( x )( mx ) x 2 + m 2 x 2 = m 1 + m 2 . This shows that for any sequence of points approaching the origin along a given line, the corresponding values of f ( −→ x ) converge—but the limit they converge to varies with the (slope of) the line , so the limit lim −→ x → −→ x f ( −→ x ) does not exist....
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 Fall '08
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 Calculus, Continuity, Derivative, Limits, Limit

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