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Unformatted text preview: D we are free to approach
xyplane. Example:
Show the Limit does not exist. Evaluate the limit along the path of the xaxis: Evaluate the limit along the path of the yaxis: 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 inclass test, maybe takehome 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/(x1) )
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.
 Spring '08
 Loukianoff,V
 Calculus

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