Its pretty easy, in each function you have
Take the limit for each one as
If the limit = 0 , then the function is continuous at (0,0)
Do you have any problems in evaluating these limits?
Consider the following functions each of which is defined on the x - y plane
f1(x) = (x-y)/(x+y) if x + y is not 0 and otherwise f1(x,y) = 0
f2(x,y) = (xy)/(x^2 + y^2) if (x,y) is not (0,0) and otherwise f2(0,0) = 0
f3(x,y) = (x^3 - y^3)/(x^2 + y^2) if (x,y) is not (0,0), and otherwise f3(0,0) is 0
Which of these is continuous
A) none B) f1 only C) f2 only D) f3 only E) all three
I know the defination of continuity for a single variable is the lim as x-> a of f(x) = f(a)
So i assume for two variables it should be the lim as (x,y) -> (a,b) of f(x,y) = f(a,b)
But I am not sure how to figure this out can someone help out please
But the lim (x,y) ->(0,0) is 0 for all them is it.. I mean Since the function is peicewise defined... If i plug in (0,0) then each one gives 0/0 Thats what i do not understand.
I understand it is in indeterminant form. But we haven't used polar coordinates. As far as algebra goes, nothing cancels when factored.
With the path rule should I approach (0,0) from the left and (0,0) from the right? I am not sure what that will do..
I see I undersand. You found two paths in which the limits are not equal so the limit does not exist because if the limit did exist, we would get the same limit no matter what path we took.
So for f2(x,y) = xy/(x^2 + y^2)
The path y = x gives me the limit = 1/2
But the path y = 0 gives me the limit = 0
This means f2(x,y) is not continuous at (0,0) either
And for f3(x,y) = x^3 - y^3/(x^2 + y^2)
the path y = x gives me the limit = 0
the path y = 1 gives me the limit = -1
So f3(x,y) is not continuous at (0,0) either. So none of them are.
is this correct?
Correct.
But you should know that this method does not work if the limit exist.
As an example:
If you take any path, the limit always will be 0
Here, you should think about another way to deal with this limit
What I want to say is that do not think in every problem that the limit does not exist and try to search for paths that have different values !