Limits of functions of 2 variables

[mit_hacker]

Homework Statement

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).

Homework 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.

The Attempt at a Solution

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

 

[Dick]

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

 

[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?

 

[Dick]

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

 

[mit_hacker]

Ohhhh ya!

I didn't see that. Thanks!

 

[Dedwardn]

Same thing different problem

Hey. how do you solve:

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

lots of thanks

 

[HallsofIvy]

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?