How to Calculate a 2-Variable Function Limit?

[maciejewski]

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

I need help with this above

any suggestions?

i don't know how to do this and have exam tomorrow :(

 

[maciejewski]

please I am short of time...

i would be very grateful

 

[flyingpig]

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]

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

 

[maciejewski]

$$x^4 \leq sin^2 y + x^4 \leq 1 + x^4$$
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 doesn't 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]

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?

 

[maciejewski]

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 don't know how to calculate these limit (d'hospital) ??

 

[LCKurtz]

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

That's correct.

but i don't know how to calculate these limit (d'hospital) ??

Have you tried anything? What happens if y → 0 first? What happens if x → 0 first?