Limits and Continuity in Multivariable Calculus

Welcome, students! Today, we're diving into the fascinating world of limits and continuity for multivariable functions. 🌟 In this lesson, you'll learn how to evaluate limits of functions with more than one variable, understand what continuity means in higher dimensions, and avoid common pitfalls along the way. By the end, you’ll have a solid grasp of these essential concepts and be ready to tackle more advanced topics in Calculus 3.

Understanding Limits of Multivariable Functions

Let's start with a quick refresher: in single-variable calculus, the limit of a function $f(x)$ as $x$ approaches a certain value gives us the function’s behavior near that point. But now, we’re working with functions of two or more variables—like $f(x, y)$ or $f(x, y, z)$. The big question: how do limits work when there's more than one variable? 🤔

Definition of a Multivariable Limit

For a function $f(x, y)$, the limit as $(x, y)$ approaches $(a, b)$ is defined as:

$\lim_{(x, y) \to (a, b)}$ f(x, y) = L

This means that as the point $(x, y)$ gets closer and closer to $(a, b)$, the value of $f(x, y)$ gets closer and closer to $L$. But here’s the tricky part: in multivariable calculus, $(x, y)$ can approach $(a, b)$ from infinitely many directions—along the $x$-axis, the $y$-axis, diagonally, or even along some curve.

For the limit to exist, $f(x, y)$ must approach the same value $L$ no matter which path we take. If the function gives different values depending on the path, the limit does not exist.

Example: A Simple Multivariable Limit

Consider the function:

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

Let’s find the limit as $(x, y) \to (0, 0)$.

Approach along the $x$-axis: Set $y = 0$. Then $f(x, 0) = \frac{2x \cdot 0}{x^2 + 0^2} = 0$.

Approach along the $y$-axis: Set $x = 0$. Then $f(0, y) = \frac{2 \cdot 0 \cdot y}{0^2 + y^2} = 0$.

Approach along the line $y = x$: Then $f(x, x) = \frac{2x \cdot x}{x^2 + x^2} = \frac{2x^2}{2x^2} = 1$.

We’ve found a problem! Approaching along the $x$-axis and $y$-axis, the limit is $0$, but approaching along $y = x$, the limit is $1$. Since the limit depends on the path, the limit does not exist. 🚩

Key Insight: Path Dependence

This example reveals a crucial idea: to check if a limit exists, we must examine multiple paths. If the function approaches the same value from every possible path, the limit exists. If not, it doesn’t.

Testing Limits: The Squeeze (Sandwich) Theorem

Sometimes, we can’t easily determine the limit by testing paths. That’s where the Squeeze Theorem can help. This theorem states that if $g(x, y) \leq f(x, y) \leq h(x, y)$ for all points near $(a, b)$, and if:

$\lim_{(x, y) \to (a, b)}$ g(x, y) = $\lim_{(x, y) \to (a, b)}$ h(x, y) = L

then:

$\lim_{(x, y) \to (a, b)}$ f(x, y) = L

Let’s apply it to the function:

$f(x, y) = x^2 y^2$

We want to find $\lim_{(x, y) \to (0, 0)} f(x, y)$.

Notice that $0 \leq x^2 y^2 \leq x^2 + y^2$ for all $(x, y)$. Also, we know that $\lim_{(x, y) \to (0, 0)} (x^2 + y^2) = 0$. And clearly, $\lim_{(x, y) \to (0, 0)} 0 = 0$. So by the Squeeze Theorem, we have:

$\lim_{(x, y) \to (0, 0)}$ x^2 y^2 = 0

This method can be a powerful tool when direct path checking gets complicated.

Polar Coordinates: A Helpful Trick

Another useful technique for evaluating limits of two-variable functions is converting to polar coordinates. Recall that in polar coordinates:

x = r $\cos$ $\theta$ \quad \text{and} \quad y = r $\sin$ $\theta$

and $r = \sqrt{x^2 + y^2}$.

Often, expressing a function in terms of $r$ and $\theta$ simplifies the limit. Let’s revisit our earlier example:

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

In polar coordinates, $x = r \cos \theta$ and $y = r \sin \theta$, so:

f(r, $\theta)$ = $\frac{2(r \cos \theta)(r \sin \theta)}{r^2}$ = $\frac{2r^2 \cos \theta \sin \theta}{r^2}$ = $2 \cos$ $\theta$ $\sin$ $\theta$

As $r \to 0$, the function becomes $2 \cos \theta \sin \theta$. Notice that this expression depends on $\theta$. As $r \to 0$, the value of $f(r, \theta)$ can vary depending on $\theta$. This confirms that the limit does not exist, as we found earlier.

Continuity of Multivariable Functions

Now that we’ve explored limits, let’s talk about continuity. In single-variable calculus, a function $f(x)$ is continuous at a point $a$ if:

$\lim_{x \to a} f(x)$ exists.
$f(a)$ is defined.
$\lim_{x \to a} f(x) = f(a)$.

For multivariable functions, the idea is the same, but we must consider points in the plane (or space) instead of just on a line.

Definition of Continuity in Two Variables

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

$\lim_{(x, y) \to (a, b)} f(x, y)$ exists.
$f(a, b)$ is defined.
$\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$.

If a function is continuous at every point in its domain, we say it’s continuous everywhere. Just like in single-variable calculus, continuity means no sudden jumps or breaks in the surface defined by $f(x, y)$.

Example: A Continuous Function

Consider the function:

$f(x, y) = x^2 + y^2$

We’ll check continuity at the point $(0, 0)$.

Limit: $\lim_{(x, y) \to (0, 0)} (x^2 + y^2) = 0$.
Value: $f(0, 0) = 0^2 + 0^2 = 0$.
Equality: $\lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0) = 0$.

So, $f(x, y)$ is continuous at $(0, 0)$. In fact, $x^2 + y^2$ is continuous everywhere, since it’s a polynomial in $x$ and $y$—and polynomials are always continuous.

Example: A Discontinuous Function

Now consider this piecewise function:

$f(x, y) = $

$\begin{cases}$

$\frac{xy}{x^2 + y^2}$ & \text{if } (x, y) $\neq$ (0, 0) \\

0 & \text{if } (x, y) = (0, 0)

$\end{cases}$

We want to check continuity at $(0, 0)$.

Limit: As we saw before, $\lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2}$ does not exist because it depends on the path.
Value: $f(0, 0) = 0$.

Since the limit doesn’t exist, $f(x, y)$ is not continuous at $(0, 0)$. This is an example of a function that’s defined at a point but not continuous there.

Common Pitfalls in Continuity

A function can be defined at a point but not continuous (like the previous example).
A function can be continuous in one variable but not in both. For instance, $f(x, y) = \frac{x}{y}$ is continuous in $x$ if $y$ is fixed and not zero, but not continuous as $(x, y) \to (0, 0)$.

Real-World Example: Temperature Distribution

Imagine a metal plate with a temperature distribution $T(x, y)$. The function $T(x, y)$ gives the temperature at each point $(x, y)$ on the plate. If $T(x, y)$ is continuous, it means there are no sudden jumps in temperature—no abrupt changes from hot to cold. On the other hand, if $T(x, y)$ is discontinuous, you might find sharp boundaries where the temperature shifts dramatically.

In real life, most physical properties—like temperature, pressure, and elevation—are described by continuous functions. That’s why understanding continuity is so important in fields like physics, engineering, and even economics.

Techniques for Proving Continuity

There are a few strategies for proving a function is continuous at a point:

Direct Substitution: For simple functions (like polynomials), you can often plug in the point and check if the limit matches the function’s value.

Factoring and Simplifying: Sometimes, functions have removable discontinuities. If you can factor and cancel terms, you might reveal a hidden continuity. For example, the function:

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

is undefined at $(x, y) = (1, 1)$, but if we factor it as:

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

and cancel $(x - y)$, we get $f(x, y) = x + y$. Now, $f(1, 1) = 2$, and the function is continuous everywhere except where $x = y$.

Using Polar Coordinates: As we saw earlier, converting to polar coordinates can help identify whether a limit exists and whether a function is continuous.

Conclusion

In this lesson, students, we explored the concepts of limits and continuity in multivariable calculus. We learned how to evaluate limits by testing different paths, using the Squeeze Theorem, and converting to polar coordinates. We also discussed how to determine if a function is continuous by checking if the limit matches the function’s value. Finally, we looked at real-world examples, like temperature distributions, to see how these concepts apply beyond the classroom. With these tools in hand, you’re ready to tackle more complex multivariable functions with confidence! 🚀

Study Notes

A limit of a multivariable function $f(x, y)$ exists if the function approaches the same value from all directions.

If the limit depends on the path taken, the limit does not exist.

Key strategies for evaluating limits:
Test along different paths (e.g., $x$-axis, $y$-axis, $y = mx$, polar coordinates).
Use the Squeeze Theorem: If $g(x, y) \leq f(x, y) \leq h(x, y)$ and the limits of $g$ and $h$ are the same, then $f$ has the same limit.
Convert to polar coordinates: $x = r \cos \theta$, $y = r \sin \theta$, and $r = \sqrt{x^2 + y^2}$.

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

$\lim_{(x, y) \to (a, b)} f(x, y)$ exists.
$f(a, b)$ is defined.
$\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$.

Polynomials in $x$ and $y$ are continuous everywhere.

Discontinuities can occur when the limit does not exist or when the function is not defined at a point.

Techniques for proving continuity:
Direct substitution (for simple functions).
Factoring and simplifying (to remove discontinuities).
Polar coordinates (to analyze behavior as $r \to 0$).

Real-world example: Temperature distribution on a surface is often modeled by a continuous function $T(x, y)$. Discontinuities would represent sudden temperature jumps, which are uncommon in physical systems.

Great job today, students! Keep practicing, and soon limits and continuity in multiple dimensions will feel like second nature. 🌟