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The concept of limit can be applied to functions of several variables. As always, we say that the limit of f(x,y)
when (x,y) approaches (x0,y0) is L when we can make f(x,y)
have values as close as we want from L provided that we
take (x,y) sufficiently close to (x0,y0).
WARNINGLimits of functions of several variables can be tricky. Usually it is much easier to show that the limit does not exist than to prove the existance of it. In order to show that there is no limit we only need to show that there are two different paths towards (x0,y0) along which f(x,y) has different limits. Here is and example: |
> #
2 2
x - y
Limit ------- = ?
(x,y) -> 0 2 2
x + y
2 2
x - y
g := (x,y) -> -------
2 2
x + y
| If we approach (0,0) along the x-axis i.e. when y=0, |
> L1 := Limit(g(x,0),x=0) = limit(g(x,0),x=0);
L1 := Limit 1 = 1
x -> 0
| But if we approach (0,0) along the line y=x, |
> L2 := Limit( g(x,x), x=0) = limit(g(x,x),x=0);
L2 := Limit 0 = 0
x -> 0
Therefore there is no limit because no matter how close we get to (0,0) we can always find points (of the form (x,0)) where
the function takes the value 1 and we can also find points
(of the form (x,x)) where the function takes the value 0.
The value of the function does not go to any limit.
ExampleDoes the limit of the function |
> f := (x,y) -> x*y^2/(x^2+y^4);
2
x y
f := (x,y) -> -------
2 4
x + y
3 2
x a
L1 := Limit ---------- = 0
x -> 0 2 4 4
x + a x
| So you may think that the limit is then 0 since no matter what straight line we take to 0 we always arrive at 0. But this is not enough for showing the the limit exists. Since there are many other ways to arrive to 0 that are not straight lines and if the limit exists ALL these other paths must also take you to the limit. For example, the path y = x^2 still gives, |
> L2 := Limit(f(x,x^2),x=0) = limit(f(x,x^2),x=0) ;
5
x
L2 := Limit ------- = 0
x -> 0 2 8
x + x
| But the path x = y^2 doesn't |
> L3 := Limit(f(y^2,y),y=0) = limit(f(y^2,y),y=0) ;
L3 := Limit 1/2 = 1/2
y -> 0
So as before the limit DOES NOT EXIST since there are different paths that lead to different limit values.
ExampleFind the limit of |
> g := (x,y) -> a*x^2*y/(x^2+y^2);
2
a x y
g := (x,y) -> -------
2 2
x + y
| Now straight lines lead to zero, |
> L1 := limit(g(x,m*x),x=0); L2 := limit(g(m*y,y),y=0);
L1 := 0
L2 := 0
| y = x^2 and x = y^2, also lead to zero, |
> L3 := limit(g(x,x^2),x=0); L4 := limit(g(y^2,y),y=0);
L3 := 0
L4 := 0
|
But we know this is not enough. We know however that if the limit exists it must be 0. The only way to be sure is to check that the
definition holds. So, let us denote by "e" (epsilon) and by "d"
(delta).
Let e > 0. We need to find d > 0 such that |
> #
2 2 1/2
(x + y ) < d
| we then have, |
> #
2
| a x y |
| ------- | < e
| 2 2 |
| x + y |
| But notice that, |
> #
2
a x |y|
------- < a |y| < a d
2 2
x + y
|
and all we need is that |
> d := e/a;
d := e/a