- #1

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## Homework Statement

Let f: R

^{2}-> R be defined by: [tex]f(x,y)= \frac{4x+y-3z}{2x-5y+2z}[/tex]

Determine if the [tex]\lim_{(x,y) \to (0,0)} f(x,y)[/tex] exists. If the limit exists prove it. If not prove that it doesn't.

## Homework Equations

## The Attempt at a Solution

I'm not sure what it means by "proving". I don't know if we need to show a rigorous proof (like epsilon & delta) or simply showing whether there is a common limit along different paths.

Here is my attempt:

As (x,y) -> (0,0) along the y-axis, x=0:

[tex]\lim_{(x,y) \to (0,0)} \frac{4x+y-3z}{2x-5y+2z} = \frac{0+y-3z}{0-5y+2z}[/tex]

As (x,y) -> (0,0) along the x-axis, y=0:

[tex]\lim_{(x,y) \to (0,0)} \frac{4x+y-3z}{2x-5y+2z} = \frac{4x+0-3z}{2x-0+2z}[/tex]

As (x,y) -> (0,0) along the line y=x:

[tex]\lim_{(x,y) \to (0,0)} \frac{4x+y-3z}{2x-5y+2z} = \frac{4x+x-3z}{2x-x+2z} = \frac{5x-3z}{x+2z}[/tex]

I'm a little confused here about the "z" & I don't know how to get rid of it...

Any help is greatly appreciated.