Limits
- Thread starter crackjoke
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What do you mean, "has limits?" Do you mean in the sense that the function is continuous at a point (x,y) if the function is equal to the limit when approached from any direction?How can i find all the points, in which the function g(x,y) = xy(xy-3)/(x-y) has limits?
thanks
Please show what you have considered - that would help clarify what you think the question is. We can't give guidance if we don't know what you are supposed to do.
What determines whether a limit exists or not?How can i find all the points, in which the function g(x,y) = xy(xy-3)/(x-y) has limits?
thanks
If g(x, y) exists at all points in an open neighborhood, does the limit exist at all points in that neighborhood?
Can you articulate, non-rigorously, a justification for your answer above?
At what points if any does g(x, y) NOT exist?
Assume that for those points a left limit exists and a right limit exists?
Does the right limit equal the left limit?
So does the limit exist or not at those points?
Yes, that makes sense informally, but it is just a different way of asking the same question. Notice that I previously asked you to think about the points where the function exists (meaning does not equal infinity)and about the limit as x and y approach such a point.
So now answer these two preliminary questions.
Where does g(x, y) exist, and where, if anywhere, does g(x, y) not exist?
As I understand the question, you need not prove that the limit of g exists at every (x, y) where g exists. It is sort of intuitively obvious that it is true although in my opinion it is not so easy to prove. If you are required to prove that, this problem suddenly got a lot harder.
Now let's consider those points where g(x, y) does not exist. Let (a, b) be one of those points. Now assume that there is a limit when y is fixed close to b and x approaches a from the right and from the left. Now show that even if both limits exist, they have opposite signs and therefore are not equal. Thus no limit exists at those points.
There may be more elegant ways to do this problem, but this brute force method will work.
How can i find all the points, in which the function g(x,y) = xy(xy-3)/(x-y) has limits?
thanks
It should be straightforward to find the set of points for which the function does NOT have a limit. What makes a function go to infinity?
You might want to treat the origin as a special case.