Functions of 3 Variables - Limits

[apw235]

Homework Statement

I am to find the limit if it exists as (x,y,z) -> (0,0,0) of
(x^2+2y^2+3z^2)/(x^2+y^2+z^2)
If it doesn't exist, show that it does not exist.

The Attempt at a Solution

I know how to do with for functions of two variables, to approach on the x axis, and y axis, and maybe even occasionally on y=x or other lines to find out if the limit exists, but I haven't been able to find out how to apply this to 3 variables.

I'm not sure if i am to take f(0,0,z), f(0,y,0), and f(x,0,0) and see if i can draw conclusions from here, or alternatively, f(0,y,z), f(x,0,z), f(x,y,0).

Thank you. (this is for a calculus 3 course, undergraduate)

 

[mighty2000]

Hello,

To find the limit in this case, you can approach the point (0,0,0) along different paths and see if the limit remains the same. This is similar to what you do for functions of two variables, but now you are working in three dimensions.

For example, you can approach the point (0,0,0) along the x-axis, y-axis, and z-axis separately. Along the x-axis, the limit would be (0^2+2(0)^2+3(0)^2)/(0^2+0^2+0^2) = 0/0, which is undefined. Similarly, along the y-axis and z-axis, the limit would also be undefined.

Next, you can approach the point (0,0,0) along different lines, such as y=x or z=x. Along y=x, the limit would be (x^2+2x^2+3x^2)/(x^2+x^2+x^2) = 6x^2/3x^2 = 2, which is a constant value. This suggests that the limit may exist, but we need to check further.

Finally, you can approach the point (0,0,0) along a parabolic path, such as x=y^2+z^2. In this case, the limit would be [(y^2+z^2)^2+2y^2+3z^2]/(y^2+z^2+y^2+z^2+z^2) = (y^4+2y^2z^2+z^4+2y^2+3z^2)/(4y^2+4z^2) = (y^2+z^2)^2/2(y^2+z^2) = (y^2+z^2)/2, which is not a constant value. This suggests that the limit does not exist.

Therefore, by approaching the point (0,0,0) along different paths, we can see that the limit does not exist for this function. I hope this helps. Good luck with your calculus 3 course!