Engineering Calculus Notes 231

# Engineering Calculus Notes 231 - it has a limit along some...

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3.1. CONTINUITY AND LIMITS 219 corresponding sequence of values of f ( x 0 ) converges to L : −→ x 0 negationslash = −→ x k −→ x 0 f ( −→ x k ) L. The same arguments that worked before show that a function converges to at most one number at any given point, so we can speak of “the” limit of the function at −→ x = −→ x 0 , denoted L = lim −→ x −→ x 0 f ( −→ x ) . For functions of one variable, we could consider “one-sided” limits, and this often helped us understand (ordinary, two-sided) limits. Of course, this idea does not really work for functions of more than one variable, since the “right” and “left” sides of a point in the plane or space don’t make much sense. We might be tempted instead to probe the limit of a function at a point in the plane by considering what happens along a line through the point: that is, we might think that a function has a limit at a point if
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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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