# proving limit doesn't exist

• Apr 26th 2009, 02:06 PM
herandi1
proving limit doesn't exist
I'm having problems with some questions on a calculus worksheet. The main one is:

'Show that the following limit does not exist:

lim x+y/x-y
(
x,y)->(0,0)

I think to prove that a limit doesn't exist you have to use the 'Two-Path test' although i'm not sure how to do this. Any help would be much appreciated. Thanks.

• Apr 26th 2009, 02:32 PM
Jester
Quote:

Originally Posted by herandi1
I'm having problems with some questions on a calculus worksheet. The main one is:

'Show that the following limit does not exist:

lim x+y/x-y

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

I think to prove that a limit doesn't exist you have to use the 'Two-Path test' although i'm not sure how to do this. Any help would be much appreciated. Thanks.

Approach along the x-axis $(y=0)$ and then approach along the y-axis $(x=0).$ You'll get two different limits.
• Apr 26th 2009, 02:57 PM
herandi1
im confused, do we need to bring in a constant with these functions, otherwise it'll still make 0/0 right?
• Apr 26th 2009, 03:10 PM
Plato
Quote:

Originally Posted by herandi1
im confused, do we need to bring in a constant with these functions, otherwise it'll still make 0/0 right?

Just say that the limit does not exist. 0/0 is meaningless.
Along the path $y=-x$ the limit is $0$.
Along the path $y=0$ the limit is $1$.
That proves that there is no limit.
• Apr 26th 2009, 11:23 PM
woof
Quote:

Originally Posted by herandi1
im confused, do we need to bring in a constant with these functions, otherwise it'll still make 0/0 right?

Simplify before you take the limit.

For y=0: $\ \ {x\over x}=1$ and then take the limit, which is 1.

For x=0: $\ \ {-y\over y}=-1$ and then take the limit, which is -1.