# 2-variable function limit

• maciejewski
In summary: Have you tried anything? What happens if y → 0 first? What happens if x → 0 first?x^4 \geq 0 And that0\leq sin^2 y \leq 1Then we know thatx^4 \leq sin^2 y + x^4 \leq 1 + x^4

#### 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 :(

please I am short of time...

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.

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

flyingpig said:
$$x^4 \leq sin^2 y + x^4 \leq 1 + x^4$$
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 :(

LCKurtz said:
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 =

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

maciejewski said:
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?

LCKurtz said:
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) ??

maciejewski said:
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?

## 1. What is a 2-variable function limit?

A 2-variable function limit is the value that a function approaches as both of its variables approach a specific point. It is similar to a regular function limit, but with two variables instead of one.

## 2. How is the limit of a 2-variable function calculated?

The limit of a 2-variable function is calculated by finding the limit along different paths approaching the point in question. If the limit is the same along all paths, then that is the value of the 2-variable function limit. If it is different, then the limit does not exist.

## 3. What is the significance of 2-variable function limits?

2-variable function limits are important in understanding the behavior of a function at a particular point. They can also be used to determine the continuity and differentiability of a function at that point.

## 4. Can a 2-variable function have a limit but not be continuous at a point?

Yes, it is possible for a 2-variable function to have a limit at a point but not be continuous at that point. This can happen when the limit exists along all paths, but there is a jump or hole at the specific point.

## 5. How can the concept of 2-variable function limits be applied in real life?

2-variable function limits are commonly used in physics and engineering to model and predict the behavior of complex systems. They can also be used in economics and finance to analyze trends and make predictions based on multiple variables.