Problem 14 - Active Calculus Multivariable

Question

For each of the following prompts, provide an example of a function of two variables with the desired properties (with justification), or explain why such a function does not exist. a. A function \(p\) that is defined at \((0,0),\) but \(\lim _{(x, y) \rightarrow(0,0)} p(x, y)\) does not exist. b. A function \(q\) that does not have a limit at \((0,0),\) but that has the same limiting value along any line \(y=m x\) as \(x \rightarrow 0\). c. A function \(r\) that is continuous at \((0,0),\) but \(\lim _{(x, y) \rightarrow(0,0)} r(x, y)\) does not exist. d. A function \(s\) such that \(\lim _{(x, x) \rightarrow(0,0)} s(x, x)=3\) and \(\lim _{(x, 2 x) \rightarrow(0,0)} s(x, 2 x)=6\) for which \(\lim _{(x, y) \rightarrow(0,0)} s(x, y)\) exists. e. A function \(t\) that is not defined at (1,1) but \(\lim _{(x, y) \rightarrow(1,1)} t(x, y)\) does exist.

Step-by-Step Solution

Answer

a. Consider the function \(p(x,y) = \dfrac{x^2 y}{x^4 + y^2}\) for \((x,y) \neq (0,0)\) and \(p(0,0)=0\). The limit as \((x,y) \rightarrow (0,0)\) does not exist, as shown by the different limiting values along paths \(y = x^2\) or \(y=\alpha x^2\). b. Consider the function \(q(x, y) = \dfrac{x^2 y}{x^2 + y^2}\) for \((x,y) \neq (0,0)\). The limit as \(x \rightarrow 0\) along any line \(y = mx\) is 0, but the overall limit at \((0,0)\) does not exist. c. This cannot happen. If a function is continuous at a point, then the limit must exist at that point. Continuity means that the function is well-defined at the point, and its limit exists and is equal to the function's value at that point. d. This cannot happen. If \(\lim _{(x, y) \rightarrow(0,0)} s(x, y)\) exists, it must have the same value for every path into \((0,0)\). e. Consider the function \(t(x, y) = \dfrac{x^2 + y^2 - 2x - 2y + 1}{(x - 1)^2 + (y - 1)^2}\) for \((x, y) \neq (1, 1)\). The function has a limit \(\lim _{(x, y) \rightarrow(1,1)} t(x, y) = 1\) but is undefined at \((1,1)\).

Step 1: a. Function defined at \((0,0)\), but limit does not exist

Consider the function \(p(x,y) = \dfrac{x^2 y}{x^4 + y^2}\) for \((x,y) \neq (0,0)\) and \(p(0,0)=0\). Let's see what happens as \((x,y) \rightarrow (0,0)\). Using polar coordinates, i.e., \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), the function depends on the transformation: \[ p(r,\theta) = \dfrac{r^3 \cos^2(\theta) \sin(\theta)}{r^4 \cos^4(\theta) + r^2 \sin^2(\theta)} \] \[ = \dfrac{r \cos^2(\theta) \sin(\theta)}{r^2 \cos^4(\theta) + \sin^2(\theta)} \] Now let's take the limit as \(r \rightarrow 0\): \[ \lim_{r \rightarrow 0} p(r,\theta) = \lim_{r \rightarrow 0} \dfrac{r \cos^2(\theta) \sin(\theta)}{r^2 \cos^4(\theta) + \sin^2(\theta)} = 0 \] However, when examining the paths \(y=x^2\) or \(y=\alpha x^2\): \[ p(x,x^2) = \dfrac{x^2 \cdot x^2}{x^4 + x^4} = \dfrac{1}{2} \] Thus, the limit does not exist as these paths do not have the same limiting value: \[ \lim _{(x, y) \rightarrow(0,0)} p(x, y) \text{ does not exist} \]

Step 2: b. Function doesn't have a limit at \((0,0)\), but same limiting value along any line \(y=mx\)

Consider the function \(q(x, y) = \dfrac{x^2 y}{x^2 + y^2}\) for \((x,y) \neq (0,0)\). Along any line \(y=mx\), we have \(q(x, mx) = \dfrac{x^3 m}{x^2 + m^2 x^2} = \dfrac{mx^3}{(1+m^2)x^2}\). Taking the limit as \(x \rightarrow 0\): \[ \lim_{x \rightarrow 0} q(x, mx) = \lim_{x \rightarrow 0} \dfrac{mx^3}{(1+m^2)x^2} = 0 \] However, the limit \(\lim _{(x, y) \rightarrow(0,0)} q(x, y)\) does not exist, since different paths have different limiting values.

Step 3: c. Continuous function at \((0,0)\), but limit does not exist

This cannot happen. If a function is continuous at a point, then the limit must exist at that point. Continuity means that the function is well-defined at the point, and its limit exists and is equal to the function's value at that point. Hence, this requirement is contradictory.

Step 4: d. Function s with different coursewise limits, but overall limit exists

This cannot happen. If \(\lim _{(x, y) \rightarrow(0,0)} s(x, y)\) exists, it must have the same value for every path into \((0,0)\). However, this prompt says that the function s must have different limiting values along \((x,x)\) and \((x,2x)\), which are contradictory requirements.

Step 5: e. Function not defined at (1,1) but limit exists

Consider the function \(t(x, y) = \dfrac{x^2 + y^2 - 2x - 2y + 1}{(x - 1)^2 + (y - 1)^2}\) for \((x, y) \neq (1, 1)\). To find the limit, we will analyze \(\lim _{(x, y) \rightarrow(1,1)} t(x, y)\). We can rewrite the function as: \[ t(x, y) = \dfrac{(x - 1)^2 + (y - 1)^2}{(x - 1)^2 + (y - 1)^2} = 1 \] for \((x, y) \neq (1, 1)\). Therefore, the function has a limit: \[ \lim _{(x, y) \rightarrow(1,1)} t(x, y) = 1\] but it is undefined at \((1,1)\).

Key Concepts

Limits of Functions

In multivariable calculus, understanding the concept of limits is crucial. A limit examines how a function behaves as the input approaches a certain point. For a function of two variables, like \( p(x, y) \), we are interested in what happens as \( (x, y) \) approaches a point, such as \( (0,0) \). Some functions can be defined at a point but have a limit that does not exist at that point. An example is the function \( p(x, y) = \frac{x^2 y}{x^4 + y^2} \), which is defined at \( (0,0) \) but has different values approaching from different paths, such as \( y = x^2 \) and \( y = \alpha x^2 \). This lack of a single approaching value demonstrates that the limit does not exist overall. Therefore, the existence of a limit requires that the function approaches the same value from every direction.

Continuity in Multivariable Functions

In the realm of multivariable calculus, continuity is a property that describes whether a function behaves smoothly at a given point. For a function of two variables to be continuous at a point, it must meet three criteria:

• The function is defined at that point.
• The limit of the function exists as it approaches that point.
• The function's value at that point equals the limit.

Attempting to create a function \( r(x, y) \) that is continuous at \( (0,0) \) but with a non-existent limit is inherently contradictory. Continuity implies a well-defined limit matching the function's value at the point. This highlights that for multivariable functions, continuity ensures no abrupt changes in value at the point in consideration.

Functions of Two Variables

Functions of two variables, such as \( f(x, y) \), are equations involving both \( x \) and \( y \). Each point in the plane described by \( (x,y) \) returns a value from the function. In exercises that involve such functions, it's common to explore how they behave near certain points or along certain paths. For instance, a function like \( q(x, y) = \frac{x^2 y}{x^2 + y^2} \) provides insight into path-dependent behavior. Testing different lines like \( y=mx \), we can see that the behavior (e.g., returning the same limit along linear paths, \( y=mx \)) might vary along different routes. Studying products or ratios of \( x \) and \( y \) enables deeper understanding of how variables interact spatially, demonstrating interesting and sometimes non-intuitive results in both behavior and limit existence.

Problem-Solving in Calculus

Solving calculus problems, particularly with multivariable functions, involves a strategic approach to understanding function behavior. It requires:

• Analyzing paths: Considers different equations like \( y=mx \) to check limit consistency along various angles.
• Checking for path dependency: Determine if limits vary with the path taken, indicating no overall limit.
• Equivalence in expressions: Rewriting expressions to reveal underlying simplicity or hidden features in function behavior.

An important part of problem-solving is recognizing when certain conditions are impossible. For example, exercises may ask for contradictory properties, such as having different path-based limits yet expecting the overall limit to exist. In such cases, explaining why the condition is unachievable solidifies understanding.Overall, calculus problem-solving is about blending analytic thinking with mathematical rigor to explore and verify function properties thoroughly.