What is the limit as (x,y) approaches (0,0) for the function x^2/(x+y)?

  • Thread starter Mingy Jongo
  • Start date
  • Tags
    Limit
In summary, the conversation discusses finding the limit as (x,y) approaches (0,0) for the function x^2/(x+y). The individual has attempted various methods but has not been able to prove the limit exists. They also discuss the possibility of the limit not existing and suggest trying a quadratic or cubic path to show this. The individual later figures out that if y=-x, the limit is actually infinity instead of zero.
  • #1
Mingy Jongo
5
0

Homework Statement



Find the limit as (x,y) approaches (0,0):

Homework Equations



x^2/(x+y)

The Attempt at a Solution



I have tried replacing y with 0, mx, x^2, etc. (and likewise with x), and all give me a limit of zero. So I tried to think of ways I could use the definition of a limit, squeeze theorem, L'Hopital's rule, etc. to prove it, but there does not appear to be a way. I'm starting to think that a limit might not even exist. Is there something elementary I'm overlooking?
 
Physics news on Phys.org
  • #2
It is a lot easier to show the limit does not exist. All you have to do is find one path towards (0,0) that produces a limit that is not zero. Maybe you could try a quadratic, or a cubic..
 
  • #3
I figured it out a while ago. If y=-x, the limit as x->0 is infinity, not zero.
 

Related to What is the limit as (x,y) approaches (0,0) for the function x^2/(x+y)?

What is a two-variable limit problem?

A two-variable limit problem is a mathematical concept that involves finding the limit of a function with two independent variables as they approach a specific point. This is typically denoted as lim (x,y) -> (a,b) f(x,y), where (a,b) represents the point of interest.

How is a two-variable limit problem solved?

To solve a two-variable limit problem, you must first approach the point of interest from various directions, such as along the x-axis, y-axis, or along a diagonal. Then, you evaluate the limit for each approach and see if they all converge to the same value. If they do, then that value is the limit of the function at the given point.

What happens if the limit does not exist for a two-variable limit problem?

If the limit does not exist for a two-variable limit problem, it means that the function does not approach a single value as the variables approach the point of interest. This could occur if the function has a discontinuity at that point or if the values of the function become increasingly larger or smaller as the variables approach the point.

What is the significance of solving a two-variable limit problem?

Solving a two-variable limit problem is essential in understanding the behavior of a function at a specific point. It can help determine if the function is continuous at that point and can also be used to find the partial derivatives of the function.

Are there any real-world applications of two-variable limit problems?

Yes, two-variable limit problems have many real-world applications, such as in physics, economics, and engineering. For example, in physics, two-variable limit problems can be used to determine the velocity and acceleration of an object at a specific time. In economics, they can be used to analyze the relationship between two variables, such as supply and demand.

Similar threads

Proving Irregularity of {(x, y, z) within R^3 : x^2 + y^2 - z^2 = 0}
    • Calculus and Beyond Homework Help
Replies
13
Views
2K
What is true for limit of f (x,y) as (x,y)→(0,1)?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Does ##\lim_{(x,y) \rightarrow (0,0)} f(x,y)## exist?
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Does this function have a limit at (0,0)?
    • Calculus and Beyond Homework Help
Replies
26
Views
2K
Two Limit exercises of functions of two variables.
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
How to solve a limit in two variables with an indeterminate form at (0,0)?
    • Calculus and Beyond Homework Help
2
Replies
35
Views
3K
Limit of a multivariable function
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Can the limit of a quotient of trig functions approach a specific value?
    • Calculus and Beyond Homework Help
Replies
2
Views
878
Let A={(x,y):x>0,x^2>y>0}. Prove for all lines through (0,0)
    • Calculus and Beyond Homework Help
Replies
3
Views
972
Confirm Limit Existence for Function f w/o Piecewise Def.
    • Calculus and Beyond Homework Help
Replies
4
Views
938

Hot Threads

Recent Insights

Back
Top