Limit of ((x^3 + (4x^2)y)/(x^2+2y^2)) as (x,y)->(0,0)

Do you know what cos(theta) and sin(theta) are as r->0?In summary, the given limit is equal to 0 and can be expressed in polar coordinates as r->0 with the help of cos(theta) and sin(theta). Canceling out the r^2 and using the properties of cos(theta) and sin(theta) in the limit will lead to the solution.
  • #1
adron
5
0

Homework Statement



Find the limit of ((x^3 + (4x^2)y)/(x^2+2y^2)) as (x,y) -> (0,0)

Homework Equations


The Attempt at a Solution



I am guessing the limit is equal to 0, and I know I have to use

0 < sqrt(x^2 + y^2) < delta where |f(x,y) - L| < epsilon

I just have no idea what to do next
I don't want anyone to give me the answer, just a point in the right direction

Sorry about the formatting, I'm quite lost when it comes to Latex.
 
Physics news on Phys.org
  • #2
Don't worry about the formatting. We can all read that. You used parentheses and everything. Express it in polar coordinates.
 
  • #3
Sorry it's been a while since I did anything to do with polar, how do I do that?
 
  • #4
x=r*cos(theta), y=r*sin(theta). You get an r^3 in the numerator and an r^2 in the denominator, yes? Cancel the r^2. Now the limit is r->0.
 

FAQ: Limit of ((x^3 + (4x^2)y)/(x^2+2y^2)) as (x,y)->(0,0)

What is a limit?

A limit is a mathematical concept that describes the behavior of a function at a specific point. It represents the value that a function approaches as its input approaches a given value.

How is the limit of a multivariable function calculated?

The limit of a multivariable function is calculated by approaching the point of interest from all possible paths and checking if the function approaches a consistent value. If it does, that value is the limit. If it approaches different values depending on the path, the limit does not exist.

What does it mean for a limit to approach zero?

When a limit approaches zero, it means that the function is getting closer and closer to zero as its input approaches a specific value. In other words, the function is becoming smaller and smaller as its input gets closer to the given value.

Why is the limit of ((x^3 + (4x^2)y)/(x^2+2y^2)) as (x,y)->(0,0) important?

This limit is important because it helps us understand the behavior of the function (x^3 + (4x^2)y)/(x^2+2y^2) as (x,y) approaches (0,0). It can also help us identify any potential issues or discontinuities in the function at that point.

What does it mean if the limit of a function does not exist?

If the limit of a function does not exist, it means that the function does not approach a consistent value as its input approaches a given value. This could be due to a discontinuity or an undefined value in the function, or because the function approaches different values depending on the path of approach.

Similar threads

Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
Replies
2
Views
916
Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection
Replies
7
Views
1K
How should I calculate the stationary value of ## S[y] ##?
Replies
2
Views
1K
Solving x+3y=4-kz & 4x-2y=5+10z: Find the Solution Here
Replies
6
Views
1K
Determining domain for C^1 function
Replies
4
Views
763
Proving Irregularity of {(x, y, z) within R^3 : x^2 + y^2 - z^2 = 0}
Replies
14
Views
2K
Question re: Limits of Integration in Cylindrical Shell Equation
Replies
24
Views
2K
Find the volume of the solid bound by the three coordinate planes
Replies
2
Views
849
Find the second derivative of the relation; ##x^2+y^4=10##
Replies
24
Views
2K
Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top