Foundations — Limits and continuity in 2D — path-dependence issue

This page is a toolbox. We open every drawer, name every tool in plain words, draw the picture it stands for, and say why the parent topic the path-dependence note cannot work without it. Read top to bottom — each item is built only from the ones above it.

1. The point in the plane: $(x,y)$

Picture: the burnt-orange dot sits at $(x,y)$. Walk $x$ units along the horizontal axis, then $y$ units up. One pair = one location.

Why the topic needs it: the whole subject asks what happens to a machine's output as the input point $(x,y)$ slides toward a special target point $(a,b)$. Without a way to name points, we can't even talk about "sliding toward."

2. The target point $(a,b)$ and the arrow $\to$

Here $(a,b)$ is a fixed spot we care about (often the origin $(0,0)$), while $(x,y)$ is the wandering point that closes in on it.

3. The function of two inputs: $f(x,y)$

Picture: think of $f$ as the height of a landscape above the flat sheet. Every ground position $(x,y)$ has a terrain height $f(x,y)$ floating above it.

Why the topic needs it: the "limit" question is "as I walk toward the target on the ground, does the terrain height settle to one number?" No two-input function, no question.

4. Distance in 2D: $\sqrt{(x-a{)}^2+(y-b{)}^2}$

This is the single most important symbol in the parent note, so we build it slowly.

How the formula is born — the right triangle. Go from $(a,b)$ to $(x,y)$ in two steps: first move horizontally a gap of $(x-a)$, then vertically a gap of $(y-b)$. These two moves are the two legs of a right triangle; the straight arrow between the points is its hypotenuse.

Pythagoras says (leg)² + (leg)² = (hypotenuse)². So

${\text{distance}}^2=(x-a{)}^2+(y-b{)}^2⟹\text{distance}=\sqrt{(x-a{)}^2+(y-b{)}^2}.$

• We square each gap so that a negative gap (moving left or down) counts the same as a positive one — length can't be negative.
• We take the square root at the end to undo the squaring and return to honest length units.

Why the topic needs it: the definition of a limit says the input must be within some small radius of the target. This formula is that "within distance." (We name that radius in item 7.)

5. Absolute value: $|\cdot |$ and the candidate value $L$

Why the topic needs it: $|f(x,y)-L|$ measures how far the output $f(x,y)$ is from the candidate answer $L$, regardless of whether it overshoots or undershoots. It is the output-side twin of the input-side distance in item 4.

6. The open disk — the picture of "close in every direction"

Picture: a filled circle (minus its edge and its exact centre). Every ray, every spiral, every zig-zag toward $(a,b)$ eventually lives inside this disk.

7. The Greek dials: $\epsilon$ (epsilon) and $\delta$ (delta)

Why the topic needs it: paths can only ever disprove a limit. To prove one exists you must win the $\epsilon$–$\delta$ game on the whole disk. These two symbols are the referee's rulebook.

8. Slope $m$ and the line $y=b+m(x-a)$

The line through $(a,b)$ with slope $m$ is $y=b+m(x-a)$. Different $m$ picks a different straight road into the target.

Why the topic needs it: the cheapest paths to test are straight lines. Sweeping $m$ over all values tests infinitely many directions at once — and if the answer still depends on $m$, the limit is already dead (Worked Example 1 in the parent note).

9. Polar coordinates: $r$ and $\theta$

Picture: the deep-teal arrow has length $r$ and swings through angle $\theta$. The tip's shadow on the horizontal axis is $r\cos \theta$ (that's $x$); its height is $r\sin \theta$ (that's $y$).

10. The squeeze idea and the bound $\le$

Why the topic needs it: it is the only honest way to conclude a limit exists (parent Worked Example 3).

11. Degree / homogeneity (the "why the slope survived" tool)

Prerequisite map

Equipment checklist

Test yourself — you should be able to answer each before reading the parent note.

Connections

• Multivariable Calculus — parent chapter this toolbox feeds
• Epsilon-Delta Definition — the $\epsilon$–$\delta$ challenge game in full
• Polar Coordinates — the $r,\theta$ language for approaching the origin
• Squeeze Theorem — the bound-based tool to prove existence
• Continuity in 1D — the 1D story this generalises
• Partial Derivatives — built from limits along axis directions
• Differentiability in 2D — needs the continuity these foundations support