de l'hospital rule for two variable function

[Lampa]

I have a simple question. Can I use de l'hospital rule in such a case

\[lim_{(x,y)->(\infty,\infty )}x^{2} -2yx + 3 = x^{2} \lim_{(x,y)->(\infty,\infty)} \frac{x^{2} -2yx + 3}{x^{2}} = \frac{[\infty]}{[\infty]}
= x^{2} \lim_{(x,y)->(\infty,\infty)} \frac{2x - 2y}{2x}
= x^{2} \lim_{(x,y)->(\infty,\infty)} \frac{2}{2}\]
or am I wrong and should do something else?

 

[tkhunny]

I have a simple question. Can I use de l'hospital rule in such a case

\[lim_{(x,y)->(\infty,\infty )}x^{2} -2yx + 3 = x^{2} \lim_{(x,y)->(\infty,\infty)} \frac{x^{2} -2yx + 3}{x^{2}} = \frac{[\infty]}{[\infty]}
= x^{2} \lim_{(x,y)->(\infty,\infty)} \frac{2x - 2y}{2x} = x^{2}
= x^{2} \lim_{(x,y)->(\infty,\infty)} \frac{2}{2}\]
or am I wrong and should do something else?

How do you decide the direction in which x and y increase without bound? Does it make difference?

Maybe really close to the x-axis? y = x/x+1
Maybe at the same speed? y = x
Maybe at a logarithmic relationship? y = log(x)
Maybe faster for y? y = e^x

Are you SURE your plan works for EVERY possible path?