Calculus III: Multivariable Functions and Limits

Moving from single-variable calculus to functions with several inputs is a pivotal leap in mathematical maturity, especially for engineering. In the real world, physical quantities—like temperature in a room, stress in a material, or pressure in a fluid—depend on multiple variables. Mastering the behavior of these multivariable functions, starting with their visualization and the foundational concepts of limits and continuity, is essential for modeling complex systems and solving advanced problems in fields from thermodynamics to machine learning.

Functions of Several Variables and Their Visualization

A function of two variables, $z=f(x,y)$, assigns a single output $z$ to each ordered pair $(x,y)$ in its domain, which is a subset of the $xy$-plane. The domain consists of all inputs for which the function's rule is defined. For example, the function $f(x,y)=\sqrt{1-x^2-y^2}$ has a domain of all points satisfying $x^2+y^2\le 1$, a closed disk of radius 1.

Graphing these functions produces surfaces in three-dimensional space. The graph of $z=f(x,y)$ is the set of all points $(x,y,z)$ satisfying the equation. Visualizing these surfaces is key. A simple example is the plane $z=2x-y+1$. A more complex one is the paraboloid $z=x^2+y^2$, which opens upward like a bowl.

Because sketching 3D surfaces by hand can be challenging, we use level curves (or contour maps) as a powerful two-dimensional alternative. A level curve is the set of points in the plane where $f(x,y)=k$, for a constant $k$. Imagine slicing the surface $z=f(x,y)$ with a horizontal plane $z=k$; the intersection projected down onto the $xy$-plane is the level curve. A collection of level curves for equally spaced $k$ values is a contour map, analogous to topographic maps used in geography. Closely spaced contours indicate a steep slope on the surface, while widely spaced contours indicate a gentle incline.

Limits and Continuity in Multiple Dimensions

The intuitive idea of a limit remains the same: as the input $(x,y)$ approaches a point $(a,b)$, does the output $f(x,y)$ approach some specific value $L$? We write this as $$\underset{(x,y)\to (a,b)}{\lim }f(x,y)=L.$$ The crucial difference from single-variable calculus is that $(x,y)$ can approach $(a,b)$ along an infinite number of paths—any curve leading into the point. For the limit to exist, the function must approach the same value $L$ along every possible path. This is a far stricter requirement.

This leads to the primary strategy for proving limits do not exist: find two different paths leading to $(a,b)$ along which the function approaches different values. For instance, consider ${\lim }_{(x,y)\to (0,0)}\frac{xy}{x^2+y^2}$. If we approach along the $x$-axis ($y=0$), the function is $0$. If we approach along the line $y=x$, the function becomes $\frac{x^2}{2x^2}=\frac{1}{2}$. Since $0\ne \frac{1}{2}$, the limit does not exist.

When a limit is suspected to exist, we must verify it formally. The epsilon-delta definition in multiple variables provides this rigorous foundation. It states: ${\lim }_{(x,y)\to (a,b)}f(x,y)=L$ if for every number $ϵ>0$ (the tolerance in the output), there exists a number $\delta >0$ such that if $0<\sqrt{(x-a{)}^2+(y-b{)}^2}<\delta$, then $|f(x,y)-L|<ϵ$. In essence, you can make $f(x,y)$ as close as you want to $L$ by restricting $(x,y)$ to be within a sufficiently small disk (not just an interval) around $(a,b)$. Proving limits with this definition is often more complex algebraically but follows the same logical structure as in single-variable calculus.

A function $f(x,y)$ is continuous at a point $(a,b)$ if three conditions hold:

1. $f(a,b)$ is defined.
2. ${\lim }_{(x,y)\to (a,b)}f(x,y)$ exists.
3. The limit equals the function value: ${\lim }_{(x,y)\to (a,b)}f(x,y)=f(a,b)$.

A function is continuous on its domain if it is continuous at every point in its domain. Most elementary functions (polynomials, rational functions, exponentials) are continuous on their domains. The continuity of multivariable functions implies their graphs have no holes, jumps, or vertical asymptotes within the domain.

The Epsilon-Delta Definition in Practice

While the formal definition is foundational, applying it to prove a specific limit builds deep understanding. Let's walk through proving ${\lim }_{(x,y)\to (1,2)}(3x+y)=5$ using epsilon-delta.

We want to show: For any $ϵ>0$, find a $\delta >0$ such that if $0<\sqrt{(x-1{)}^2+(y-2{)}^2}<\delta$, then $|(3x+y)-5|<ϵ$.

First, simplify the expression we need to bound: $|(3x+y)-5|=|3x+y-5|=|3(x-1)+(y-2)|$. Using the triangle inequality: $|3(x-1)+(y-2)|\le 3|x-1|+|y-2|$.

Now, note that for any point $(x,y)$ within a distance $\delta$ of $(1,2)$, we have $|x-1|\le \sqrt{(x-1{)}^2+(y-2{)}^2}<\delta$ and similarly $|y-2|<\delta$. Therefore, $3|x-1|+|y-2|<3\delta +\delta =4\delta$.

We want this to be less than $ϵ$. So if we choose $\delta =ϵ/4$, then whenever $\sqrt{(x-1{)}^2+(y-2{)}^2}<\delta$, we have $|(3x+y)-5|<4\delta =4(ϵ/4)=ϵ$. This completes the proof. The key was to connect the distance in the output ($|f(x,y)-L|$) to the geometric distance in the input ($\sqrt{(x-a{)}^2+(y-b{)}^2}$).

Common Pitfalls

1. Assuming Single-Variable Limit Rules Apply Directly to Paths: A common error is to check only the limits along the $x$- and $y$-axes and conclude the limit exists. As shown, a function can agree along these two lines but differ along a more complicated path like $y=x^2$. You must consider all paths, or provide an epsilon-delta proof.

Correction: Use the axes as an initial test, but if they agree, you cannot conclude the limit exists. You must either use the epsilon-delta definition or algebraic manipulation (e.g., conversion to polar coordinates) to prove it, or test more pathological paths to search for a counterexample.

1. Misunderstanding the Domain in Continuity: Students often incorrectly state a function is "discontinuous" at points outside its domain. For example, $f(x,y)=1/(x^2+y^2)$ is not discontinuous at $(0,0)$; it is simply undefined there. Continuity is only discussed for points in the domain of the function.

Correction: First, identify the domain of the function. A point of discontinuity must be in the domain where the definition of continuity (limit equals function value) fails.

1. Incorrect Visualization from Contour Maps: It's easy to mistake a contour map for the graph of the surface itself. A circle on a contour map represents a constant height, not a "hill." A set of concentric circles could represent either a bowl-shaped paraboloid (if values increase inward) or an inverted bowl (if values decrease inward).

Correction: Always note the labeled value $k$ for each contour. Trace a path on the map and ask: if the $k$ values are increasing along my path, am I walking uphill or downhill on the actual surface?

1. Algebraic Errors with Epsilon-Delta Proofs: The most frequent mistake is failing to properly bound the expression $|f(x,y)-L|$ in terms of the geometric distance $\delta$. Attempting to solve for $\delta$ directly from $|f(x,y)-L|<ϵ$ is usually impossible.

Correction: Use inequalities (like the triangle inequality) to overestimate $|f(x,y)-L|$ by a simpler expression involving $|x-a|$ and $|y-b|$, which are each less than $\delta$. Then choose $\delta$ to make this overestimation less than $ϵ$.

Summary

• Multivariable functions map points in the plane (or higher-dimensional space) to real numbers. Their graphs are surfaces in 3D, and level curves (contour maps) provide a crucial 2D tool for visualizing these surfaces and their rates of change.
• The limit ${\lim }_{(x,y)\to (a,b)}f(x,y)=L$ exists only if $f(x,y)$ approaches $L$ along every possible path of approach to $(a,b)$. Finding two paths with different limits is a standard method to prove a limit does not exist.
• The rigorous epsilon-delta definition formalizes the limit concept by guaranteeing you can control the output's precision ($ϵ$) by restricting the input to a small disk of radius $\delta$ around the target point.
• A function is continuous at a point $(a,b)$ if the limit exists there and equals the function's value. Understanding continuity and limits is the bedrock for the subsequent calculus of multivariable functions: differentiation and integration.