Question

Limit of a function of three variables

Answer 1

\[\lim_{(x,y,z) \rightarrow (0,0,0)} \frac{ x^2+2y^2+3z^2 }{ xy+yz+xz }\]We are trying to show that this limit does not exist. An idea of mine is to approach the point (0,0,0) along parameterized curves. would it work to follow the limit along the curve \[r(t)=<t,t,t>\] and the curve \[s(t)=<t^2,t,t>\]where r(t) and s(t) are vector-valued functions. Since the point (0,0,0) is on the curve traced out by the functions, they are valid for the limit, right? Using those parameterizations, I found that the limit it 2 along the curve r(t) and 5 along the curve s(t). Since these limits are not equal, the limit does not exist. Is this a proper way of showing this?

Answer 2

Actually, I got 5/2 as the point is approach along s(t).

Answer 3

you can also use the comparison of (x,x,x)->(0,0,0) and (x,2x,x)->(0,0,0)
i think that's simpler

Answer 4

So your limits could be parameterized b <t,t,t> and <t,2t,t>? I just use parameterized functions because the instructor prefers that method.

Answer 5

it gives the same result though, so yeah you could do it my way or the professors way

Answer 6

All right, fantastic. Thanks for the help.

Answer 7

welcome!