Limits of functions of 2 variables

1. Oct 22, 2007

mit_hacker

1. The problem statement, all variables and given/known data

By considering different paths of approach, show that the function below has no limit as (x,y) ---> (0,0).

f(x,y) = - x / sqrt(x^2 + y^2).

2. Relevant equations

This is the problem! I do not know the different techniques to find the limits of functions of more than one variable. My book only shows examples of cases where you can get the answer by substituting y=mx or y=kx^2.

3. The attempt at a solution

I tried the above substitutions but they don't work.

2. Oct 22, 2007

Dick

Why do you say y=mx doesn't work? You don't even get a limit for m=0.

3. Oct 22, 2007

mit_hacker

You do don't you?

When we substitute y=mx, we get -1/sqrt(1+m^2). So when m=0, the limit will be -1 won't it?

4. Oct 22, 2007

Dick

Depends on whether x is positive or negative. sqrt(x^2)=abs(x).

5. Oct 22, 2007

mit_hacker

Ohhhh ya!!

I didn't see that. Thanks!!

6. Mar 12, 2008

Dedwardn

Same thing different problem

Hey. how do you solve:

lim(x,y)-->(0,pi/2) ( x/cos(y) )

lots of thanks

7. Mar 12, 2008

HallsofIvy

Staff Emeritus
Have you noticed that these problems do NOT ask you to find the limit but to show that the limit does not exist? That is much simpler. Here, what limit do you get if you first let x go to 0, then let y go to $\pi/2$? What limit do you get if you first let y go to $\pi/2$, then let x go to 0? What does that tell you?