Proving Limits of Functions of Two Variables: A Comprehensive Approach

In summary, the speakers discuss how to prove the limit of a function of two variables as (x,y) approaches (0,0). They consider using approaches along all lines y = cx and show that the limiting value is independent of the value of c. However, this is not sufficient as there are examples where the limit does not exist. They also consider approaching along some line and proving the limit is Q, but this also does not guarantee the limit exists for all lines. Instead, the use of polar coordinates is suggested as it allows for a more comprehensive analysis of all possible directions and can show that the limit exists and is equal to 0. Overall, the speakers realize the importance of considering all possible approaches in order to prove the existence
  • #1
radiogaga35
34
0
Hello

I've only recently begun studying (in math, anyway) functions of two variables, so forgive me if my terminology is vague or incorrect (or if I am completely misguided!).

Suppose I want to find the limit of f(x,y) as (x,y) --> (0,0). Now, I know that the limit will exist (other conditions satisfied) only if f(x,y) --> A as (x,y) --> (0,0) from ALL directions.

To prove that the limit = Q, would it thus be sufficient to be able to show a.) that for approaches along ALL lines y = cx (c some real number), the limiting value f(x,y) will be independent of the value of c? (and since all curves through the origin can be approximated in this vicinity by a tangent line through the origin, surely this implies that all possible directions are accounted for) And then b.) that if the limiting value can be found to be Q for approach along SOME line, then by a.), the limit must exist and = Q?

Is this acceptable? Thanks for your help.
 
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  • #2
radiogaga35 said:
Hello

I've only recently begun studying (in math, anyway) functions of two variables, so forgive me if my terminology is vague or incorrect (or if I am completely misguided!).

Suppose I want to find the limit of f(x,y) as (x,y) --> (0,0). Now, I know that the limit will exist (other conditions satisfied) only if f(x,y) --> A as (x,y) --> (0,0) from ALL directions.

To prove that the limit = Q, would it thus be sufficient to be able to show a.) that for approaches along ALL lines y = cx (c some real number), the limiting value f(x,y) will be independent of the value of c? (and since all curves through the origin can be approximated in this vicinity by a tangent line through the origin, surely this implies that all possible directions are accounted for)
No, that is not sufficient. For example, if
[tex]f(x,y)= \frac{x^2y}{x^4+ y^4}[/tex]
for (x,y) not (0, 0), then the limit of f(x,y), as (x,y) goes to (0,0) along any straight line, is 0. But the limit of f(x,y), as (x,y) approaches (0,0), along the parabola, y= x2, is 1/2, so there are points arbitrarily close to (0, 0) such that f(x,y) is NOT close to 0 as well as point arbitrarily close to (0, 0) such that f(x,y) is NOT close to 1/2. That function has no limit at (0,0).

(That example is from Salas, Hille, and Etgen's Calculus text.)

And then b.) that if the limiting value can be found to be Q for approach along SOME line, then by a.), the limit must exist and = Q?
Well, no, absolutely not! There are much simpler examples where approaching (0, 0) along two different lines give different values and so the limit itself does not exist.
For example,
[tex]f(x,y)= \frac{xy+ y^3}{x^2+ y^2}[/itex]
obviously gives limit 0 as you approach (0, 0) along the x and y axes but limit 2/5 as you approach along the line y= x.
(Again, that example is from Salas, Hille, and Etgen.)

Is this acceptable? Thanks for your help.
No, it is not acceptable. You can't prove that you get the same limit approach by every possible path- there are just too many possibilities.

In my opinion, the best thing to do is to convert to polar coordinates. That way, the "closeness" to (0, 0) is determined entirely by the variable r- if you can show that the limit, as r goes to 0, is independent of [itex]\theta[/itex], then the limit exists and is that value. For example, suppose
[tex]f(x,y)= \frac{3x^3}{x^2+ y^}[/tex]
as long as (x, y) is not (0,0). We can look at as many lines or curves through (0, 0) as we wish and show that the limit is 0 but we can do all possible curves so that doesn't prove that the limit exists.

Changing to polar coordinates, [itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so
[tex]f(x,y)= f(r,\theta)=\frac{3r^3 cos^3(\theta)}{r^2cos^2(\theta)+ r^2sin^2(\theta)}= \frac{r^3 cos^3(\theta)}{r^2}= r cos(\theta)}[/tex]
which goes to 0 as r goes to 0 no matter what [itex]\theta[/itex] is.

That tells us that, close enough to (0, 0), the value of f(x,y) is close to 0 so the limit is 0.
 
  • #3
Concerning a), the set of all lines of the form y = cx does not include the line consisting of the y-axis so not all possible directions are accounted for. And why would the value of c not matter?

Concerning b), just because for some line the limit is Q doesn't mean that for other lines the limit will be Q as well.
 
  • #4
Ok, I see where I went wrong. Thank you for your help! HallsOfIvy, the polar co-ordinate transformation seems like a very good approach. Thanks a lot!
 

What are limits in functions of 2 variables?

Limits in functions of 2 variables refer to the value that a function approaches as the input variables approach a particular point in the domain. It can also be thought of as the output of a function at a specific point in the domain.

How do you calculate limits in functions of 2 variables?

To calculate limits in functions of 2 variables, you can use the same methods as in single-variable functions. You can take the limit along a specific path, such as approaching the point from the x-axis or y-axis, or you can use algebraic methods to manipulate the function and find the limit.

What is the significance of limits in functions of 2 variables?

Limits in functions of 2 variables are important in studying the behavior of a function at a specific point in the domain. They can help determine the continuity, differentiability, and convergence of a function, and they are crucial in understanding the behavior of surfaces and curves in three-dimensional space.

What are the common misconceptions about limits in functions of 2 variables?

One common misconception about limits in functions of 2 variables is that they only exist at a specific point in the domain. In reality, a limit can exist at a point even if the function is not defined at that point. Another misconception is that limits are only used to find the output of a function at a specific point, when in fact they can also be used to analyze the behavior of a function as a whole.

How are limits in functions of 2 variables used in real-world applications?

Limits in functions of 2 variables have various real-world applications, such as in economics, physics, and engineering. For example, they can be used to model and analyze the relationship between two variables, such as cost and production in economics, or velocity and time in physics. They are also important in optimization problems, where finding the maximum or minimum value of a function involves taking limits.

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