# How to find limit of multivariable function?

• Puchinita5
In summary, the first homework question could not be solved, while the second one could be solved using the squeeze theorem.
Puchinita5

## Homework Statement

Hi. I'm going to provide two homework examples. The first one I got correctly. But the second one, I used the same method as the first question but I got the answer incorrect. I'm really confused.

The first question was
$$\lim_{(x,y)\to(0,0)}\frac{(xycosy)}{3x^2+y^2}$$
f(x,0)=0
f(0,y)=0
for y=x , f(x,x)= (1/4)cosx

since the three limits are not equal, the limit does not exist!

Okay now the second question:

$$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}$$

I again did:
f(x,0)=0
f(0,y)=0
$$f(x,x)=\frac{x^2}{x\sqrt{2}}$$

since they aren't the same, limit does not exist. I THOUGHT.
I looked in the solutions manual, and it says "We can see that the limit along any line through (0,0) is 0, as well as along other paths through (0,0) such as x=y^2. So we suspect that the limit exists and equals 0; we use the Squeeze Theorem to prove our assertion.

$$0\leq\left | (\frac{xy}{\sqrt{x^2+y^2}})\right |\leq \left |x \right | since \left |y\right |\leq \sqrt(x^2+y^2) and \left |x\right | \to \0 \ as (x,y) \to \(0,0).\ so\ the\ limit\ equals\ 0.$$

I guess I don't understand how you choose the method to solve for the limit. Is there something about the function that tells you what to do? Why is f(x,x) not an option?

I mean, I guess it makes sense because I'm sure you can't just plug in anything or at some point you will get something that won't equal the same thing. But how do you know what to do? How would I have known to use the squeeze theorem?

I'm so confused.
[

this always happens to me...the SECOND i post something on this site i realize something.

I am supposed to plug in a zero into the (x^2)/(x2^.5) aren't I?
which would also make it equal to 0.

In which case...can someone just explain to me the logic behind the squeeze theorem in this example? It still confuses me.

And if you test out different functions, such as f(x,x) or f(x,0) etc... and get the same limit for all of them, do you always need to use the squeeze theorem to verify that it is indeed the limit?

Limits of functions of two variables can be very tricky. The problem is that there is not a standard approach that you can count on work any problem. So, given a problem where you don't know whether the limit exists, the usual thing is to check to see if you can get different answers along different paths. If you can, that is usually pretty easy and it settles the problem giving that the limit doesn't exist.

If you keep getting the same answer along many different paths, that may be because the limit does exist and is equal to that common number. Proving it is then where the difficulty usually arises. If you suspect, for example, that the limit is 0, then you can try to overestimate the function with something easier that does go to 0. That is the idea of the squeeze principle. You will learn a few techniques along the way to try on such problems after which they won't seem quite so intimidating. Don't get too hung up on it if you are having trouble with these at first. Experience and seeing more examples will help. And your course will soon move on to other more enjoyable topics.

## 1. What is a multivariable function?

A multivariable function is a mathematical function that takes in more than one input variable and produces an output value. This is different from a single variable function, which only takes in one input variable.

## 2. Why do we need to find the limit of a multivariable function?

Finding the limit of a multivariable function allows us to understand the behavior of the function as the input variables approach a certain value. This is important in many real-world applications, such as optimization problems in engineering and physics.

## 3. How do you find the limit of a multivariable function?

To find the limit of a multivariable function, you must first determine the direction in which the input variables are approaching the limit point. Then, you can evaluate the function along various paths approaching the limit point and see if the values approach a single value. If they do, that value is the limit of the function at that point.

## 4. What are some common techniques for finding limits of multivariable functions?

Some common techniques for finding limits of multivariable functions include using algebraic manipulation, factoring, and the Squeeze Theorem. You can also use L'Hospital's rule, which involves taking the derivatives of the function to simplify the expression and then finding the limit.

## 5. Are there any special cases when finding limits of multivariable functions?

Yes, there are some special cases when finding limits of multivariable functions. One example is when the function has a discontinuity at the limit point, in which case the limit does not exist. Another special case is when the function approaches different values along different paths, also resulting in a non-existent limit.

Replies
10
Views
1K
Replies
5
Views
775
Replies
4
Views
870
Replies
10
Views
1K
Replies
1
Views
751
Replies
4
Views
547
Replies
6
Views
2K
Replies
6
Views
784
Replies
8
Views
1K
Replies
8
Views
755