Limits and Continuity Multi-Variable Functions: Topics of Partial Derivatives

Overview of Functions With Multiple Variables

If you’ve been following in our blog, you may have found our discussion of vector calculus before arriving here. If not, it may prove useful to start with this topic; you may find it here. Nevertheless, limits and continuity are fundamental to conceptualizing the various aspects of calculus, and that consideration holds true in the case of partial derivatives. Our previous article in this series addressed the nature of functions with multiple variables and the methodologies to model these functions. So, in this article, we address limits and continuity as they pertain to functions with multiple variables. This is essential in your understanding of partial derivatives holistically.

General Understanding of Limits

In our discussion of vector calculus, we presented the case of limits and continuity in vectors. For functions of a single variable, the limit is traditionally defined as:

\text{If f(x) is a function defined on an interval x=a, then the}\newline \text{limit of the function f(x) is defined as:}\newline \lim\limits_{x\to a}f(x)=L

We will persist to address how limits for functions with multiple variables differ from this definition.

Limits in Functions with Multiple Variables

In a function defined by multiple variables, one of the form z=f(x,y), there are not too many significant differences for identifying the limits. For this function, as the ordered pair (x,y) approaches some point, the function ‘f’ will approach some value. We may thus define the limit for functions with multiple variables as follows:

\text{Let f be a function of two variables, (x,y). Let the}\newline \text{domain of these variables D be arbitrarily close to (a,b)}\newline \text{If this is true, then the limit of f(x,y) as (x,y) approaches}\newline \text{(a,b) is L, and we model this as:}\newline \lim\limits_{(x,y)\to (a,b)}f(x,y)=L

Let us consider what this means in a practical example. Consider the function which follows:

f(x,y)=\frac{sin(x^2+y^2)}{x^2+y^2}

With this function, we might seek to define the limit of this function as the values of ‘x’ and ‘y’ approach some value. Let us attempt to identify this limit as the ordered pair (x,y) approaches (0,0). This limit will take the form

\lim\limits_{(x,y)\to (0,0)}\frac{sin(x^2+y^2)}{x^2+y^2}

If we make the values of ‘x’ and ‘y’ infinitesimally close to 0, we see that the function approaches the value 1 such that:

\lim\limits_{(x,y)\to (0,0)}\frac{sin(x^2+y^2)}{x^2+y^2}=1

Proving the Limit of Functions With Multiple Variables

There is a slight caveat to our definition of the limit which allows us to prove it is true. We may say that the limit for a function f(x,y) is equal to L if:

\text{For every number}\enspace \epsilon \enspace \text{> 0 there is a corresponding}\newline \text{number}\enspace \delta \enspace \text{such that if (x,y)} \in \text{D and}\newline 0<\sqrt{(x-a)^2+(y-b)^2}, \text{then:}\newline |f(x,y)-L|<\epsilon

Let us consider exactly what this statement claims. Firstly, suppose that the function f(x,y) produces a graph of a surface ‘S’. The value

\sqrt{(x-a)^2+(y-b)^2}

reflects the distance between the points (x,y) and (a,b). If D is the domain of the function ‘f’, then the value δ represents a region of the domain encompassing the points (x,y) and (a,b). Therefore, one of the pre-requisites for our limit implies that the distance between these two points must be less than the magnitude of δ, or otherwise the two points must both be within the domain.

Now, consider the value:

|f(x,y)-L|

This value represents the difference between f(x,y) and L. It allows us to see how close the function has come to approaching the limit. Now, the value ε can be thought of as a margin of error, or a range which f(x,y) must be within for the limit L to be true. We can use these statements to geometrically model the proof.

Example of Proving Limit of a Function

Let us attempt to prove that the limit of a function exists. Let us do so by considering the limit of the following function:

\lim\limits_{(x,y) \to (0,0)}\frac{3x^2y}{x^2+y^2}

Let us first assume that ε>0. Then, we want to find the value of δ such that:

\text{if}\enspace 0<\sqrt{x^2+y^2}<δ, \text{then:} \newline \vert\frac{3x^2y}{x^2+y^2}\vert<ε

Using this we might write an equivalence statement such that:

\frac{3x^2|y|}{x^2+y^2}<3|y|=3\sqrt{y^2}<3\sqrt{x^2+y^2}

If this is true, then let us choose a value of delta such that δ=ε/3. With this:

\text{if}\enspace 0<\sqrt{x^2+y^2}<\delta \enspace\text{then:}\newline \vert\frac{3x^2y}{x^2+y^2}\vert<3\sqrt{x^2+y^2}<3\delta = \epsilon

From this, we can compute the limit of the function such that:

\lim\limits_{(x,y) \to (0,0)}\frac{3x^2y}{x^2+y^2}=0

On the Subject of Continuity

To understand the application of continuity to functions of multiple variables, conceptualization of the fundamental principles remains of essential importance. In our discussion of vector calculus, continuous functions visibly appear to be smooth, without break or sharp directional changes. However, we have available a formal definition for functions of multiple variables, which relies upon limits.

Suppose we have a function ‘f’ comprised of two variables ‘x’ and ‘y’ such that the function takes the form f(x,y). Then we might say that our function f(x,y) is continuous at the point (a,b) if:

\lim\limits_{(x,y) \to (a,b)}f(x,y)=f(a,b)

This definition implies that the function is continuous on the point (a,b) if the limit is equal to the value of the function at (a,b). Therefore, we can actually say that the function ‘f’ is entirely continuous on the interval D if the function is continuous at all points (a,b) in D. In another sense, this just means that for minimal changes in (x,y), continuous functions tend to exhibit minimal changes in f(x,y).

Example of Evaluating Continuity

Let’s suppose we have at hand a function of the following form:

f(x,y)=\frac{x^2-y^2}{x^2+y^2}

With this function, consider a scenario wherein we might desire to evaluate its continuity. How might we do this? Well, let us begin by considering the domain of the function. Because this function is a rational function, if the function has a denominator of 0, then it will not be defined. Well, if we have a point (0,0), this presupposition would be true. Therefore, we can say that the function is discontinuous at the point (0,0) because it is continuous on its defined domain which is:

D=\{(x,y)\enspace |\enspace (x,y) \not = (0,0)\}

Future Directions

This article has discussed at length the design and use of limits and continuity as they pertain to higher order functions. While this subject may be mundane, it is critical to the understanding of partial derivatives. Our next article in the series will discuss the taking of partial derivatives directly, so having a solid foundation will be helpful to understanding this critical concept. External sources applicable to this discussion can be found here. Before moving forward, consider taking a look at other articles in the series, found below:

Modeling Functions of Multiple Variables

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