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A function without a limit, although limits exist along all lines

Define f(x,y) by

displaymath531

Along any line through the origin, f approaches zero. Indeed, any line through (0,0) may be parametrized by x=ta, y=tb for some constants a and b, not both zero. As t goes to 0, tex2html_wrap_inline547 , which goes to 0 as t goes to zero for any a and b. However, tex2html_wrap_inline555 does not exist: if you approach the origin along the parabola tex2html_wrap_inline557 , you note that tex2html_wrap_inline559 is constantly 1/2, which doesn't go to zero.

Here's a picture, where we're looking along the positive x axis towards the origin. What happens as you approach along any ray which is above the x axis is that you go up the ridge which is above tex2html_wrap_inline557 and then slide down the other side as you hit the origin. However, if you come in on tex2html_wrap_inline557 , you stay on top of the ridge the whole way, and get a limit of 1/2. (If you come in on a ray below the x axis, you go down into the trough which is below tex2html_wrap_inline575 and then up on the other side until you hit the origin.)

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Tom Vogel
Mon May 5 12:53:33 CDT 1997