Limit of a Multivariable functionNEED HELP

In summary, when finding the limit of [6*(x^3)*(y^2)] / [2*(x^4) + (y^4)] as (x,y) approaches (0,0), converting to polar coordinates and simplifying the expression yields a limit of 0. However, care must be taken to ensure that the correct power of r is used in the numerator.
  • #1
*Helix*
18
0

Homework Statement



Asked to find the limit of [6*(x^3)*(y^2)] / [2*(x^4) + (y^4)] as (x,y) is approaching (0,0)

Homework Equations



lim, as (x,y) ---> (0,0), of [6*(x^3)*(y^2)] / [2*(x^4) + (y^4)]

x = rcos(theta); y = rsin(theta)

The Attempt at a Solution



Tried numerous times with polar equations x = rcos(theta); y = rsin(theta)

came up with lim r -->0 [6r^2( (costheta)^3(sintheta)^2))] / [r^4(2*(costheta)^4 + (sintheta)^4]

D.N.E? ...I think its zero... can't prove it though...HELP!
 
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  • #2
Welcome to PF, *Helix*.


*Helix* said:
came up with lim r -->0 [6r^2( (costheta)^3(sintheta)^2))] / [r^4(2*(costheta)^4 + (sintheta)^4]


There's your algebra error. The numerator has the wrong power of r.
 
  • #3
ohh..so the function becomes : [6r(costheta)^3(sintheta)^2] /[2(costheta)^4 + (sintheta)^4] and as r is approaching zero ---> 0/(a number that will never be zero) is zero
 

1. What is the definition of a limit for a multivariable function?

The limit of a multivariable function is defined as the value that a function approaches as the independent variables approach a specific point in the domain. It can also be seen as the output of the function at that point.

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

To find the limit of a multivariable function, you can use the limit laws and techniques such as substitution, factoring, and rationalization. You can also use graphs and tables to estimate the limit.

3. What is the importance of finding the limit of a multivariable function?

The limit of a multivariable function helps us understand the behavior of the function at a specific point in its domain. It is also useful in determining the continuity, differentiability, and local behavior of the function.

4. Can the limit of a multivariable function exist but not be equal to the function's value at that point?

Yes, the limit of a multivariable function can exist but not be equal to the function's value at a specific point. This usually occurs when the function is not continuous at that point.

5. How do we determine if a multivariable function has a limit at a specific point?

To determine if a multivariable function has a limit at a specific point, we can use the epsilon-delta definition of a limit or check if the limit exists using the limit laws and techniques. We can also use the concept of one-sided limits to check if the limit exists from different directions.

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