We lose graphical meaning beyond that dimension in 2d

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Unformatted text preview: D we are free to approach xy-plane. Example: Show the Limit does not exist. Evaluate the limit along the path of the x-axis: Evaluate the limit along the path of the y-axis: Thus the limit does not exist. along any path in the Example: Along the path Along the path Along the path Does this mean the limit exists and is four fifths? We only looked at 3 paths – ALL paths must be looked at which is impossible so we need a Theorem to help us out. We could also use the formal limit definition. (we will not be required to use the formal definition on an in-class test, maybe take-home test) Theorem 2.1 Suppose that Proof: (pg 941) The theorem doesn’t seem to help since now we’re just stuck with showing that . How is that any better? Instead of looking for a g(x,y) we really will be looking for a g(x) or g(y) since we know how to find those limits in ONE variable. If we can’t find that, then we are stuck using the formal definition. For the above example: We already suspect the limit L = 4/5. So I’m stuck – so I will use the formal definition. But before I do that, I should graph the surface to see if I can visually see that it has a limit! Another Example using Theorem 2.1 Exercise 27 Along the path (notice how easy it was to find a function of one variable for g) THUS Exercise 28 Along the path THUS Continuity: Calc 101A Version (2D) is continuous at when 1) is defined 2) 3) when 1) is defined 2) 3) Calc 101C Version (3D) is continuous at Find ALL points where a given function is Continuous. 101A Review: What conditions give us a discontinuity? 1) Holes (y = x/x or y = x2 / x) 2) Asymptotes ( y = 1/(x-1) ) 3) Undefined or Restricted Domain ( y = ln(x) ) 4) Piecewise defined functions ( y = 1 for x 0 and y = -1 for x 0 ) 1, 2, and 3 are easy to identify. Piecewise defined functions are tricky since you will most likely have to take a limit. Example: (by the way, in 3D Calc C, Holes become entire curves) Notice this function is still undefined when x = 0 and y disc...
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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.

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