# 2-variable function limit

$lim_{(x,y)\rightarrow(0,0)}\frac{y^{3}}{x^{4}+sin^{2}y}$

I need help with this above

any suggestions?

i dont know how to do this and have exam tomorrow :(

i would be very grateful

I'll give you my take on this

Note that

$$x^4 \geq 0$$

And that

$$0\leq sin^2 y \leq 1$$

Then we know that

$$x^4 \leq sin^2 y + x^4 \leq 1 + x^4$$

Also,all of this is just to tell you that you may have to use the squeeze theorem to solve this.

LCKurtz
Homework Helper
Gold Member
Try a couple of different paths, like letting y → 0 first or x → 0 first to see if perchance they are different.

$$x^4 \leq sin^2 y + x^4 \leq 1 + x^4$$
Also,all of this is just to tell you that you may have to use the squeeze theorem to solve this.

thanks for a tip, but I've already figured this and it probably doesnt take me any step further :(

Try a couple of different paths, like letting y → 0 first or x → 0 first to see if perchance they are different.

Do you mean calculating $lim_{x\rightarrow0}(lim_{y\rightarrow0}A)$ , where A =

$\frac{y^{3}}{x^{4}+sin^{2}y}$

LCKurtz
Homework Helper
Gold Member
Do you mean calculating $lim_{x\rightarrow0}(lim_{y\rightarrow0}A)$ , where A =

$\frac{y^{3}}{x^{4}+sin^ {2}y}$

Yes. And the reverse order too. What can you conclude if they come out not equal to each other?

Yes. And the reverse order too. What can you conclude if they come out not equal to each other?

That the limit as (x,y)->(0,0) does not exist?

but i dont know how to calculate these limit (d'hospital) ??

LCKurtz