Limits of Multivariable Functions

The central idea

The limit $\lim_{\mathbf x\to\mathbf a}f(\mathbf x)=L$ means that for every $\varepsilon>0$ there is $\delta>0$ such that $0<\|\mathbf x-\mathbf a\|<\delta$ and $\mathbf x$ in the domain imply $|f(\mathbf x)-L|<\varepsilon$. The value must be the same along every path and sequence of approach.

Definitions, hypotheses, and notation

Polar substitution $x=r\cos\theta$, $y=r\sin\theta$ is useful when the transformed expression has a factor $r^k$ times a function of angle bounded independently of $\theta$. Then $r^k\to0$ uniformly for $k>0$. If the angular factor becomes unbounded or the result depends on $\theta$, further analysis or a counterexample is needed.

Iterated limits, taking $x$ then $y$, are not equivalent to the joint limit. They may agree even when a diagonal path disagrees. The norm-based definition is the authoritative statement because it controls all nearby points simultaneously.

Conceptual meaning

In more than one dimension there are infinitely many approach geometries. Two paths with different limits disprove existence, but agreement along several paths does not prove a limit. A proof needs uniform control, often through distance $r$ or a squeeze bound.

A dependable method and decision rule

1. Try continuity laws and direct substitution first.
2. If an indeterminate form remains, test strategically chosen lines or curves for nonexistence.
3. For existence, seek an absolute bound depending only on $r=\|\mathbf x-\mathbf a\|$.
4. Use polar or spherical coordinates only with an angle-uniform bound.
5. State whether the result is a proof, a disproof, or merely diagnostic evidence.

Fully worked example

Graphical or geometric meaning

For the first function, every point over a shrinking disk lies in a shrinking vertical band. For the second, two ridges aimed at the same base point retain different heights, preventing a single limiting value.

Common mistakes and why they fail

Verification and reasonableness checks

• Convert bounds into powers of radial distance.
• Test at least one nonlinear path when searching for counterexamples.
• Use numerical surfaces only as guidance, not proof.

Path tests disprove; bounds prove

A multivariable limit must be the same along every path in the domain. Two paths giving different values prove nonexistence. Agreement along lines, parabolas, or many sampled directions never proves existence because infinitely many approaches remain. To prove a limit, rewrite or bound the expression by a function of distance $r=\|\mathbf x-\mathbf a\|$ that tends to zero. Polar coordinates help near the origin when angular factors remain uniformly bounded, but dependence on $\theta$ may instead reveal failure. Iterated limits are not equivalent to the full limit, though unequal iterated limits disprove it. State the domain of approach when a denominator or boundary restricts available points.

Practice

1. Find $\lim_{(x,y)\to0}(x^2+y^2)$.
2. Does $x^2/(x^2+y^2)$ have a limit at zero?
3. What does one path with a different limit prove?

Connections and next steps

• Continuity in Several Variables
• Functions of Several Variables
• Directional Derivatives