Functions of 3 Variables - Limits

• apw235
In summary, to find the limit in this case, you can approach the point (0,0,0) along different paths and see if the limit remains the same. By approaching the point along different paths, we can see that the limit does not exist for this function.
apw235

Homework Statement

I am to find the limit if it exists as (x,y,z) -> (0,0,0) of
(x^2+2y^2+3z^2)/(x^2+y^2+z^2)
If it doesn't exist, show that it does not exist.

The Attempt at a Solution

I know how to do with for functions of two variables, to approach on the x axis, and y axis, and maybe even occasionally on y=x or other lines to find out if the limit exists, but I haven't been able to find out how to apply this to 3 variables.

I'm not sure if i am to take f(0,0,z), f(0,y,0), and f(x,0,0) and see if i can draw conclusions from here, or alternatively, f(0,y,z), f(x,0,z), f(x,y,0).

Thank you. (this is for a calculus 3 course, undergraduate)

Hello,

To find the limit in this case, you can approach the point (0,0,0) along different paths and see if the limit remains the same. This is similar to what you do for functions of two variables, but now you are working in three dimensions.

For example, you can approach the point (0,0,0) along the x-axis, y-axis, and z-axis separately. Along the x-axis, the limit would be (0^2+2(0)^2+3(0)^2)/(0^2+0^2+0^2) = 0/0, which is undefined. Similarly, along the y-axis and z-axis, the limit would also be undefined.

Next, you can approach the point (0,0,0) along different lines, such as y=x or z=x. Along y=x, the limit would be (x^2+2x^2+3x^2)/(x^2+x^2+x^2) = 6x^2/3x^2 = 2, which is a constant value. This suggests that the limit may exist, but we need to check further.

Finally, you can approach the point (0,0,0) along a parabolic path, such as x=y^2+z^2. In this case, the limit would be [(y^2+z^2)^2+2y^2+3z^2]/(y^2+z^2+y^2+z^2+z^2) = (y^4+2y^2z^2+z^4+2y^2+3z^2)/(4y^2+4z^2) = (y^2+z^2)^2/2(y^2+z^2) = (y^2+z^2)/2, which is not a constant value. This suggests that the limit does not exist.

Therefore, by approaching the point (0,0,0) along different paths, we can see that the limit does not exist for this function. I hope this helps. Good luck with your calculus 3 course!

What is the definition of a limit for a function of 3 variables?

The limit of a function of 3 variables is the value that the function approaches as the inputs of the variables get closer and closer to a specific point in 3-dimensional space.

How is the limit of a function of 3 variables calculated?

The limit is calculated by evaluating the function at various points surrounding the given point and observing the trend of the function's values as the points get closer and closer to the given point.

Can a function of 3 variables have multiple limits?

Yes, a function of 3 variables can have multiple limits if the function approaches different values when the inputs approach a given point from different directions.

What is the difference between a finite and infinite limit of a function of 3 variables?

A finite limit means that the function approaches a specific value as the inputs get closer to the given point. An infinite limit means that the function grows or decreases without bound as the inputs approach the given point.

How do the properties of limits for functions of 3 variables compare to those of functions of 1 or 2 variables?

The properties of limits for functions of 3 variables are similar to those of functions of 1 or 2 variables, but they involve considering the behavior of the function in 3-dimensional space rather than just on a 2-dimensional plane.

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