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

Common mistakes and why they fail