Problem 14 - The Calculus with Analytic Geometry

Question

In Exercises 13 through 16, prove that \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) exists. \(f(x, y)=\frac{x^{3}+y^{3}}{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

\[\text{Therefore, } \lim _{(x, y) \rightarrow (0,0)} f(x, y) = 0\]

Step 1: Determine the Limit Path

To prove that a limit exists, evaluate the function along different paths approaching the origin. In this case, consider the paths where x = 0 and y = 0 separately.

Step 2: Evaluate the Limit Along the Path y = 0

Substitute y = 0 into the function: oindent\[f(x, 0) = \frac{x^{3}+0^{3}}{x^{2}+0^{2}} = \frac{x^{3}}{x^{2}} = x\]oindentAs \(x\) approaches 0, \(f(x,0) = x\) approaches 0.

Step 3: Evaluate the Limit Along the Path x = 0

Now, substitute x = 0 into the function: oindent\[f(0, y) = \frac{0^{3}+y^{3}}{0^{2}+y^{2}} = \frac{y^{3}}{y^{2}} = y\]oindentAs \(y\) approaches 0, \(f(0,y) = y\) approaches 0.

Step 4: Evaluate the Limit Along the Path y = x

Consider the path y = x. Substitute y = x into the function: oindent\[f(x, x) = \frac{x^{3}+x^{3}}{x^{2}+x^{2}} = \frac{2x^{3}}{2x^{2}} = x\]oindentAs \(x\) approaches 0, \(f(x,x) = x\) approaches 0.

Step 5: Conclude the General Limit

Since along all paths chosen (y = 0, x = 0, and y = x), the function approaches the same limit (0), conclude that the limit is consistent regardless of path.oindent\[\text{Therefore, } \lim _{(x, y) \rightarrow (0,0)} f(x, y) = 0\]

Key Concepts

Limit Paths

When working with multivariable limits, understanding the concept of limit paths is crucial. These are specific routes or equations you use to approach a particular point (in this case, the origin where \(x = 0\) and \(y = 0\)).
By evaluating the function along different paths, we can test if the limit is the same in multiple directions.
For instance:

• Substitute \(y = 0\) to see how the function behaves when moving along the x-axis.
• Substitute \(x = 0\) to analyze the function's behavior along the y-axis.
• Use \(y = x\) to understand the function when moving along the line where y equals x.

If the function approaches the same value along all these paths, it suggests that the limit at that point exists.

Evaluating Limits

Evaluating limits in multivariable functions involves substituting different paths into the function. This helps confirm whether the same limit value is approached consistently.
Let’s take the function from the exercise as an example: \(f(x, y) = \frac{x^{3}+y^{3}}{x^{2}+y^{2}}\)
We start by evaluating along the path \(y=0\):
\[f(x, 0) = \frac{x^{3}}{x^{2}} = x\] As \(x\) approaches 0, so does \(f(x, 0) \rightarrow 0\).
Next, try the path \(x=0\):
\[f(0, y) = \frac{y^{3}}{y^{2}} = y\] As \(y\) approaches 0, \(f(0, y) \rightarrow 0\).
Finally, consider the path \(y=x\):
\[f(x, x) = \frac{2x^{3}}{2x^{2}} = x\] Similarly, as \(x\) approaches 0, \(f(x, x) \rightarrow 0\).
This consistent approach to zero from different paths strongly indicates that the limit is indeed zero.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions with multiple variables.
Instead of dealing with simple linear limits, we handle functions that change in two or more dimensions.
A fundamental idea here is limits, essential for understanding continuity and differentiability in higher dimensions.
For functions of two variables, limits depend on all directions from which a point can be approached. So, comprehensive testing along multiple paths is required.
Returning to our function \(f(x, y)\), verifying limits from different paths ensures that the function behavior is consistent in its multidimensional domain.
If the limit varies based on the approach path, it indicates the limit does not exist at that point.

Approaching Origin

Approaching the origin (0,0) in a multivariable context means evaluating the function as both variables (x and y) simultaneously move towards zero.
This is not as straightforward as single-variable limits and requires checking multiple paths to ensure consistency.
Consider the paths used in the problem to test the limit as (x, y) approaches (0, 0):

• Path \(y = 0\) moves along the x-axis.
• Path \(x = 0\) moves along the y-axis.
• Path \(y = x\) moves along the line where both coordinates are equal.

In each case, substituting these paths in \(\frac{x^{3}+y^{3}}{x^{2}+y^{2}}\) consistently led to zero, verifying that \( \text{lim}_{(x, y) \to (0,0)} f(x, y) = 0 \).
This thorough approach ensures the limit exists and is the same regardless of the path taken.