How would I approach a question such as:

find the limits:

1-lim{ (x^4 + y^4) / (x^2+ y^2)} as (x,y) tends to (0,0)

2-lim{ xy^2 / (x^2 + y^4)} as (x,y) tends to (0,0)

Thanks

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- Apr 17th 2009, 03:50 AMbigdoggyLimit of Vector Valued Function
How would I approach a question such as:

find the limits:

1-lim{ (x^4 + y^4) / (x^2+ y^2)} as (x,y) tends to (0,0)

2-lim{ xy^2 / (x^2 + y^4)} as (x,y) tends to (0,0)

Thanks - Apr 17th 2009, 04:16 AMrubix
edit: nvm what i just said.

polar coordinate that Twig showed is the way to go. It drops you down from two variables (x, y) to one (r). - Apr 17th 2009, 04:23 AMTwig
Hi

1st one:

Use polar coordinates,

Gives us:

And here

2nd one:

If we let , then the expression is already zero, which approaches zero.

But if we let

What does this tell you? - Apr 17th 2009, 04:46 AMbigdoggy
Hi

It tells me there is no limit at the origin since there are points close to the origin which the function takes the value 1/2 and not 0 ?

Why use polar coordinates on the first one?

Also, are there any pointers you can advise on how to approach these questions, i.e. finding the value of a limit at a point(for vector valued fns)? - Apr 17th 2009, 04:58 AMTwig
hi

I used polar coordinates because I find it convenient and relatively easy here to show that no matter what angle we approach by, the function tends to zero.

I´m not that familiar with vector valued functions to be honest, I have only done these types of limits when .

But I would imagnine that both limits have to exist for a vector to approach some point so to speak.