Finding limits of 2 varible functions

[tnutty]

Homework Statement

Find the limit, if it exist, otherwise show that it doesn't exist.

lim F
(x,y) -- > (0,0) ,

where F = xycos(y) / (3x^2 + y^2)


I let T(x,y) = F

Then evaluated the limit as (x,y) - >(x,0) = 0;
The same goes for the vertical line.

The answer is it does not exist, But I can't seem to prove it.

 

[slider142]

Let y = tx where t is an arbitrary non-zero constant. This is approaching the point from a line of finite non-zero slope.

 

[tnutty]

Let me try :

F(x,tx) = x(tx)*cos(tx) / (3x^2 + (tx)^2)

= tx^2*cos(tx) / (3x^2 + t^2*x^2)
=

tx^2*cos(tx)
-------------
x^2(3 + t^2)

=

t*cos(tx)
-------------
(3 + t^2)

I know that cos(tx) <= 1

so :

t*cos(tx) t
----------- <= -------
(3 + t^2) 3 + t^2not sure what to do next?

 

[slider142]

That's it. 1/(3 + t²) is never 0, and approaching along the x or y-axis yields 0. Since the two limits are not equal, the limit does not exist. Recall the definition of the limit.

 

[tnutty]

OH right, Thanks