Re: find limit (function in two variables)
- From: Randy Poe <poespam-trap@xxxxxxxxx>
- Date: Mon, 3 Dec 2007 14:00:53 -0800 (PST)
On Dec 3, 4:39 pm, "Carl R." <solrac...@xxxxxxxxxxx> wrote:
Hello , can you please help
define f(x,y) = x^2y^2/sqrt(x^2+y^2) if (x,y) is different from (0,0)
and f(x,y) = (0,0) if (x,y) = (0,0)
How you do find limit f(x,y) as (x,y)-> (0,0)?
I tried to let y=0 then along the x axis the limit is 0, similarly for
x=0, thus
the limit is 0? therefore f is continuous at 0?
No, that is not enough to conclude that f is continuous
at 0. You have merely established continuity along
two lines. To prove continuity you need to show that the
limit is the same for all possible sequences of
points approaching the origin.
I would suggest using the epsilon-delta
definition. For any epsilon > 0, can you find
delta > 0 such that for all | (x,y) | < delta,
|f(x,y)| < epsilon? This is more general than what
you did, because it considers ALL points within
radius delta.
Hint: The denominator is | (x,y) |, and
| (x,y) | < delta implies certain things about
|x| and |y| and thus the numerator.
- Randy
.
- References:
- find limit (function in two variables)
- From: Carl R.
- find limit (function in two variables)
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