What is the type of discontinuity at the origin for the function 2x^y/x^4+y^2?

[kazthehack]

1.)

Homework Statement

Sketch the domain of f(x,y,z) = ln(25-x^2-y^2-z^2) and determine it's range.

Homework Equations

N/a

The Attempt at a Solution

Im sure the domain is x^2+y^2+z^2 < 25 or (-infinity, 25)
Then the range is (0,+infinity.)
my problem is how would i sketch it.. i was thinking of a sphere with dotted outline at 5,5,5

2.

Homework Statement

Specify the type of discontinuity at the origin of the following functions.

2x^y/x^4+y^2

Homework Equations

N/a

The Attempt at a Solution

i am not sure how would i find the limit of f(x,y) as (x,y)->(0,0) so i can prove that it is either essential or removable discontinuity..

any help would be great thanks! :D

 

[Mark44]

1.)

Homework Statement

Sketch the domain of f(x,y,z) = ln(25-x^2-y^2-z^2) and determine it's range.

Homework Equations

N/a

The Attempt at a Solution

Im sure the domain is x^2+y^2+z^2 < 25 or (-infinity, 25)

The domain is {(x, y, z) | x^2+y^2+z^2 < 25 }. This domain is a subset of three-dimensional space, not an interval as you show it.

Then the range is (0,+infinity.)

No. Think about the largest and smallest values that ln(25-x^2-y^2-z^2) can attain. Sure, it will always be > 0, but there is a finite value that it is always less than.

my problem is how would i sketch it.. i was thinking of a sphere with dotted outline at 5,5,5

You're on the right track with the sphere, as the domain is the interior of some sphere, but the point (5, 5, 5) is outside the sphere. I don't understand what you're trying to say with "dotted outline at 5,5,5."

2.

Homework Statement

Specify the type of discontinuity at the origin of the following functions.

2x^y/x^4+y^2

Please write this more carefully so I can clearly see what's in the denominator. Most people on this forum would interpret this as 2(x^y/x⁴) + y², but I suspect that isn't what you mean.

Homework Equations

N/a

The Attempt at a Solution

i am not sure how would i find the limit of f(x,y) as (x,y)->(0,0) so i can prove that it is either essential or removable discontinuity..

any help would be great thanks! :D

 

[kazthehack]

So as for the first question the graph would be a sphere that is shaded inside? and then the outline of the sphere would be dotted to indicate that it wouldn't reach the radius of 5.

As for the 2nd one ...
the equation was 2x^2y/(x^4+y^2)

 

[Mark44]

So as for the first question the graph would be a sphere that is shaded inside? and then the outline of the sphere would be dotted to indicate that it wouldn't reach the radius of 5.

Do you mean the graph of the domain of the function or the graph of the function? The domain of your function is the interior of a sphere centered at (0, 0, 0) and of radius 5.

I haven't discussed the graph of the function, other than to say something about its domain and range.

As for the 2nd one ...
the equation was 2x^2y/(x^4+y^2)

Well, that's what I though you meant, but I wanted to get you to write it so that it would be generally understandable.

For the limit as (x, y) --> (0, 0) of this function to exist, it must exist independent of the path taken. Try approaching (0, 0) along different lines (y = kx) and different curves, and see if they come out the same. For different curves, you might try y = x², y = x³, and so on.

 

[kazthehack]

No. Think about the largest and smallest values that ln(25-x^2-y^2-z^2) can attain. Sure, it will always be > 0, but there is a finite value that it is always less than.

Can you clarify this?

 

[Dick]

Can you clarify this?

25-x^2-y^2-z^2=25-(x^2+y^2+z^2). What's the largest value that can ever be?