Limits & Continuity: Multivariable — Limits and continuity in multiple variables (conceptual) - Multivariable Calculus

SECTION 1

Historical Context & Motivation

In single-variable calculus you learned to ask, "What value does f(x) approach as x gets close to some number?" That idea is powerful, but the real world rarely depends on just one variable. Temperature on a metal plate depends on both its x- and y-coordinates. Air pressure depends on latitude, longitude, and altitude. Once mathematicians realized they needed to handle functions of several variables, they had to figure out what limits and continuity should mean when a point can be approached from infinitely many directions instead of just two (left and right).

1715 • Brook Taylor's Series in Several Variables

Brook Taylor extended power-series ideas, hinting that functions of more than one variable needed their own careful treatment of convergence and limiting behavior.

1821 • Cauchy Formalizes Limits

Augustin-Louis Cauchy published rigorous ε–δ definitions for single-variable limits, creating the logical foundation that would later be extended to multiple dimensions.

1861 • Weierstrass Perfects Rigor

Karl Weierstrass refined Cauchy's ε–δ framework and applied it broadly, including to functions of several variables, giving mathematicians the modern standard of proof for limits.

1900s • Multivariable Calculus Matures

By the early 20th century, multivariable limits and continuity became standard topics in analysis textbooks, fueled by applications in physics, engineering, and economics.

The central question this lesson addresses is deceptively simple: when you can approach a point from every possible direction and along every possible curved path, how do you determine whether a function actually settles on one value? And if it does, what does it mean for the function to be continuous there?

SECTION 2

Core Principles & Definitions

Before diving into calculations, let's nail down the foundational ideas that make multivariable limits different from the single-variable version you already know. In one variable, "approaching a point" means sliding along a number line from the left or the right. In two or more variables, you can approach from infinitely many directions—north, south-east, along a parabola, spiraling inward—and the limit must be the same no matter which path you take.

Path Independence

A multivariable limit exists only if the function approaches the same value along every possible path toward the point. Finding even one path that gives a different value proves the limit does not exist.

Open Disks Replace Open Intervals

In one variable, neighborhoods are open intervals (a, b). In two variables, neighborhoods are open disks of radius δ centered at the point. In three variables, they become open balls.

Continuity = Limit Equals Value

A function f(x, y) is continuous at (a, b) when three things hold: f(a, b) is defined, the limit of f as (x, y) → (a, b) exists, and that limit equals f(a, b).

Two Paths Test (Disproof Tool)

The quickest way to show a limit does not exist is the two-paths test: find two different paths to the point that yield different limiting values.

Squeeze Theorem Generalizes

The Squeeze Theorem still works: if you can trap f(x, y) between two functions that both approach L, then the limit of f is also L.

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KEY TAKEAWAY

Think of a multivariable limit like a concert stage surrounded by an audience on all sides. In single-variable calculus the audience sits in two lines (left and right). Now imagine the audience surrounding the stage in a full circle—or even a sphere. The performer (the limit value) must sound exactly the same to every single listener, no matter where they sit. If even one person hears a different note, the limit doesn't exist.

SECTION 3

Visual Explanation — Approaching a Point from Many Paths

The diagram below shows several paths converging on the origin in the xy-plane. Each colored arrow represents a different direction or curve along which (x, y) approaches (0, 0). For a limit to exist, the function's output must approach the same value along every one of these paths (and every other path you can imagine).

Six sample paths approach the origin: along the x-axis, the y-axis, the lines y = x and y = −x, and the parabolas y = x² and y = −x². The dashed circle represents the δ-disk neighborhood around the point.

Notice the dashed circle—this is the δ-disk. In one variable you check whether f(x) is within ε of L whenever x is within δ of the target. Here you check whether f(x, y) is within ε of L whenever the point (x, y) falls inside a disk of radius δ centered at the target point. The fundamental logic is the same, but the geometry jumps from a line segment to a disk (or, in three variables, a ball).

SECTION 4

Mathematical Framework

Let's translate the intuitive ideas from the previous sections into precise mathematical language. Although you may not be asked to write formal proofs at the high-school level, understanding what the notation means will help you reason about limits correctly.

FORMAL DEFINITION OF A MULTIVARIABLE LIMIT

lim₍ₓ,ᵧ₎→₍ₐ,ᵦ₎ f(x, y) = L

This means: for every ε > 0 there exists a δ > 0 such that |f(x, y) − L| < ε whenever 0 < √((x − a)² + (y − b)²) < δ. The expression √((x − a)² + (y − b)²) is the distance from (x, y) to the target point (a, b).

DISTANCE IN THE PLANE

d = √((x − a)² + (y − b)²)

This is the standard Euclidean distance formula from geometry. The δ-disk is the set of all points whose distance from (a, b) is less than δ.

CONTINUITY CONDITION

f is continuous at (a, b) ⟺ lim₍ₓ,ᵧ₎→₍ₐ,ᵦ₎ f(x, y) = f(a, b)

Three things must be true simultaneously: (1) f(a, b) is defined, (2) the limit exists, and (3) the limit equals f(a, b). If any one of these fails, f is discontinuous at that point.

ℹ

Why Can't We Just Check Left and Right?

In single-variable calculus, a limit exists when the left-hand limit and right-hand limit are equal. In multiple variables there is no finite checklist of directions. You must account for approaches along straight lines at every angle, along parabolas, spirals, and any other curve. This is why disproving a limit is often easier than proving one—one bad path is enough to disprove, but you'd need to check infinitely many paths to prove.

SECTION 5

Path Analysis — When Limits Fail

One of the trickiest situations in multivariable calculus occurs when every straight-line path gives the same limiting value, yet the limit still does not exist because some curved path gives a different answer. The classic example is the function f(x, y) = xy² / (x² + y⁴). Let's see why path analysis matters so much.

The function f(x, y) = xy / (x² + y²) yields a different limiting value for each straight-line path y = mx. Along the x-axis (m = 0) the limit is 0, along y = x (m = 1) it is ½, and along y = −x it is −½. Since at least two paths disagree, the limit does not exist.

The diagram illustrates a powerful technique. When you substitute a path like y = mx into f(x, y), the function reduces to a single-variable expression in x. If the resulting limit depends on the slope m, then different directions give different answers, and the multivariable limit does not exist. This strategy is called the two-paths test, even though you often try several paths to build intuition before settling on two that disagree.

⚠

Watch Out — All Straight Lines Agreeing Isn't Enough!

For the function f(x, y) = xy² / (x² + y⁴), every line y = mx gives limit 0, yet the parabolic path x = y² gives limit ½. This proves the limit doesn't exist even though all straight lines agree. Always consider curved paths when you suspect trouble!

SECTION 6

Worked Example

Let's work through two complete examples: one where the limit exists and one where it does not.

Example A — Limit Exists

Find lim₍ₓ,ᵧ₎→₍₀,₀₎ (x² y) / (x² + y²)

1
Step 1 — Try the x-axis path (y = 0)

Substitute y = 0: f(x, 0) = (x² × 0) / (x² + 0) = 0 for all x ≠ 0.

Limit along x-axis = 0
2
Step 2 — Try the y-axis path (x = 0)

Substitute x = 0: f(0, y) = (0 × y) / (0 + y²) = 0 for all y ≠ 0.

Limit along y-axis = 0
3
Step 3 — Try the line y = x

Substitute y = x: f(x, x) = x³ / (x² + x²) = x³ / (2x²) = x/2, which → 0 as x → 0.

Limit along y = x is 0
4
Step 4 — Confirm with the Squeeze Theorem

Since y² / (x² + y²) ≤ 1 for all (x, y), we have |x² y / (x² + y²)| ≤ |x² y| / x² = |y| (when x ≠ 0). More generally, |f(x, y)| ≤ |y| because x² / (x² + y²) ≤ 1. As (x, y) → (0, 0), |y| → 0, so by the Squeeze Theorem the expression is squeezed to 0.

By the Squeeze Theorem, the limit is 0

Example B — Limit Does Not Exist

Show that lim₍ₓ,ᵧ₎→₍₀,₀₎ (x² − y²) / (x² + y²) does not exist

1
Step 1 — Try the x-axis path (y = 0)

f(x, 0) = x² / x² = 1 for all x ≠ 0.

Limit along x-axis = 1
2
Step 2 — Try the y-axis path (x = 0)

f(0, y) = −y² / y² = −1 for all y ≠ 0.

Limit along y-axis = −1
3
Step 3 — Conclude

Two paths give two different values (1 and −1). Because a multivariable limit must be path-independent, this is enough to conclude that the limit does not exist.

The limit does not exist (DNE)

SECTION 7

Single-Variable vs. Multivariable — Strengths & Limitations

Understanding where single-variable intuition still works—and where it breaks down—is crucial for mastering multivariable limits. The table below highlights the most important contrasts.

Feature	Single-Variable f(x)	Multivariable f(x, y)
Approach directions	Two: left and right	Infinitely many: every direction, line, curve
Neighborhood shape	Open interval (a − δ, a + δ)	Open disk of radius δ (or ball in 3-D)
Proving limit DNE	Show left ≠ right limit	Find any two paths with different limits
Proving limit exists	Show left = right (or direct ε–δ)	Squeeze Theorem, polar coordinates, or ε–δ proof
Continuity definition	lim f(x) = f(a)	lim f(x, y) = f(a, b); same three conditions
Algebraic shortcuts	Factor, rationalize, L'Hôpital's Rule	Same, plus convert to polar; L'Hôpital does not directly apply

Key differences between single-variable and multivariable limits

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KEY TAKEAWAY

In single-variable calculus, checking two directions (left and right) is enough to prove a limit exists. In multivariable calculus, checking finitely many paths is never enough to prove existence. You need a global argument like the Squeeze Theorem or an ε–δ proof. However, a single pair of disagreeing paths is always enough to disprove existence.

SECTION 8

Connection to Advanced Theory

The concepts in this lesson form the bedrock for more advanced topics in multivariable calculus. Once you're comfortable with limits and continuity, several powerful ideas open up.

This Lesson's Concept	Where It Leads
Multivariable limit definition	Partial derivatives — defined as limits of difference quotients in each variable separately
Continuity of f(x, y)	Differentiability — a stronger condition than having partial derivatives; requires continuity of partials
δ-disk neighborhoods	Topology of ℝⁿ — open sets, closed sets, compact sets, and boundary points
Path independence of limits	Directional derivatives & gradients — studying the rate of change in any direction, not just along axes
Squeeze Theorem in 2-D	Polar/spherical coordinate techniques — converting to r and θ often simplifies squeeze arguments dramatically

How today's ideas connect to advanced multivariable calculus

A particularly useful extension is converting to polar coordinates (x = r cos θ, y = r sin θ) when evaluating limits at the origin. In polar form, the distance from the origin is simply r, and you need to show that the expression approaches L as r → 0 regardless of θ. If the result depends on θ, the limit does not exist. This technique bridges nicely into the gradient and directional derivative concepts you will encounter next.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain in your own words why checking the limit along the x-axis and y-axis alone is not enough to prove that a multivariable limit exists. What additional requirement must be satisfied?

Show answer

In single-variable calculus, the approach is restricted to a number line, so checking left and right covers all possibilities. In multivariable calculus, a point can be approached from infinitely many directions—straight lines at every angle, parabolas, spirals, and arbitrary curves. Checking just the x-axis and y-axis tests only two of these infinitely many paths. The limit exists only when the function approaches the same value along every path, so two paths are never sufficient for proof (though they can be sufficient for disproof if they disagree).

Question 2: BASIC CALCULATION

Evaluate lim₍ₓ,ᵧ₎→₍₁,₂₎ (3x + y²). Is the function f(x, y) = 3x + y² continuous at (1, 2)?

Show answer

Since f(x, y) = 3x + y² is a polynomial in x and y, it is continuous everywhere. To evaluate the limit, simply substitute: f(1, 2) = 3(1) + 2² = 3 + 4 = 7. Because polynomials are continuous at every point, the limit equals the function value. Therefore lim₍ₓ,ᵧ₎→₍₁,₂₎ (3x + y²) = 7, and yes, f is continuous at (1, 2).

Question 3: INTERMEDIATE

Determine whether lim₍ₓ,ᵧ₎→₍₀,₀₎ (2xy) / (x² + y²) exists. Support your answer by testing at least two paths.

Show answer

Along the x-axis (y = 0): f(x, 0) = 0 / x² = 0, so the limit is 0. Along the line y = x: f(x, x) = 2x² / (x² + x²) = 2x² / (2x²) = 1, so the limit is 1. Since two different paths give values 0 and 1, the limit does not exist. Note that this function is essentially 2 × [xy / (x² + y²)], which is the classic path-dependent example scaled by a factor of 2. Any pair of lines y = m₁x and y = m₂x with m₁ ≠ m₂ generally produces different limiting values, confirming path dependence.

Question 4: APPLIED

A sensor measures temperature T(x, y) = (x² y) / (x² + y²) across a metal plate, with the origin representing a tiny hole where the sensor can't take a reading. Does the temperature approach a well-defined value near the hole? Use the Squeeze Theorem to justify your answer.

Show answer

We need to evaluate lim₍ₓ,ᵧ₎→₍₀,₀₎ (x² y) / (x² + y²). Observe that x² / (x² + y²) ≤ 1 for all (x, y) ≠ (0, 0) because the numerator is at most the denominator. Therefore |T(x, y)| = |x² y / (x² + y²)| ≤ |y|. Since |y| → 0 as (x, y) → (0, 0), the Squeeze Theorem gives lim T(x, y) = 0. Yes, the temperature approaches the well-defined value of 0 near the hole, even though the sensor cannot directly measure it there. If we define T(0, 0) = 0, the function becomes continuous at the origin.

Question 5: CRITICAL THINKING

Consider f(x, y) = (x y²) / (x² + y⁴). Show that the limit along every straight line y = mx through the origin equals 0, then find a curved path that gives a different limiting value. What does this tell you about the overall limit?

Show answer

Along y = mx: f(x, mx) = x(m²x²) / (x² + m⁴x⁴) = m²x³ / (x²(1 + m⁴x²)) = m²x / (1 + m⁴x²) → 0 as x → 0, for any fixed m. So every straight line gives limit 0. Now try the parabolic path x = y²: f(y², y) = (y²)(y²) / ((y²)² + y⁴) = y⁴ / (y⁴ + y⁴) = y⁴ / (2y⁴) = 1/2 for all y ≠ 0. Along this curved path the limit is 1/2, not 0. Since different paths yield different values, the overall limit does not exist. This example powerfully demonstrates that straight-line tests alone can be misleading—you must consider curved paths too.

SUMMARY

Lesson Summary

Summary

A multivariable limit extends the single-variable concept by requiring that f(x, y) approach the same value L along every possible path toward the target point. Instead of checking only left and right, you must consider infinitely many directions and curves. The two-paths test is a powerful disproof tool: if any two paths yield different limiting values, the limit does not exist. However, agreement along all straight lines is not enough to prove existence—curved paths must be checked as well.

To prove a limit exists, use a global argument such as the Squeeze Theorem or a formal ε–δ proof. Neighborhoods change from intervals to δ-disks (or balls in higher dimensions). A function f(x, y) is continuous at (a, b) when f(a, b) is defined, the limit exists, and the limit equals f(a, b)—the same three-part definition from single-variable calculus. These ideas are the foundation for partial derivatives, differentiability, and the gradient that you'll study next.