Limits and continuity in 2D — path-dependence issue: Visual walkthrough

We assume you know only: a point $(x, y)$ is a spot on a flat map, and a function $f (x, y)$ gives that spot a height. Everything else — the disk, the ray, the slope, the polar trick — we earn on the way.

Step 1 — What a 2D limit is actually asking

WHAT. Picture a flat map (the $x y$ -plane) and above every spot a height $f (x, y)$ . We hover the origin and ask: does the height near it approach one fixed value?

WHY the word "disk". In 1D you approach a point from the left road or the right road — only two ways. On a map you can walk in from any direction and along any curve. The honest way to capture "close to the origin from every side at once" is a small filled circle — a disk — of radius $δ$ (a tiny distance we get to shrink). Look at the figure: the shaded blue disk is all the points within distance $δ$ of the origin. The limit rule demands the height be near $L$ on the whole disk, not on a couple of roads.

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

Here $δ$ (Greek letter "delta") is just a name for "how close is close enough", and $x^{2} + y^{2}$ is the straight-line distance from $(x, y)$ to the origin — the length of the arrow drawn on the figure. See Epsilon-Delta Definition for the full precision version.

Step 2 — Our test function, and reading it as a height

We study the parent's canonical villain: $f (x, y) = height at spot (x, y) \frac{x y}{x ^{2} + y ^{2}} .$

$x y$ (the top) is positive when $x, y$ share a sign, negative when they differ, zero on either axis.
$x^{2} + y^{2}$ (the bottom) is the squared distance to the origin — always positive except at the origin itself, where it is $0$ (so $f$ is undefined exactly at the point we are limiting toward — that is why it is interesting).

WHAT. The figure colours the map by height: warm where $f > 0$ , cool where $f < 0$ .

WHY. Before any algebra, seeing the colouring already whispers the answer. Notice the colour never fades to one shade as you spiral toward the centre — near the origin you find warm and cool packed together. A well-behaved limit would show one uniform colour swallowing the centre. This one does not.

Step 3 — The cheapest paths: the two axes

WHAT. Walk in along the $x$ -axis. On that road $y = 0$ , so $f (x, 0) = \frac{x \cdot 0 y}{x ^{2} + y ^{2} 0} = \frac{0}{x ^{2}} = 0.$ Every step of the walk reads height $0$ ; the limit along this road is $0$ .

Now the $y$ -axis, where $x = 0$ : $f (0, y) = \frac{0 x \cdot y}{x ^{2} 0 + y ^{2}} = \frac{0}{y ^{2}} = 0.$ Also $0$ .

WHY these first. They are the cheapest substitutions — one variable becomes zero and the whole thing collapses to a 1D calculation. The picture shows both axis-roads glowing at height $0$ all the way to the centre.

Step 4 — All the slanted lines at once

WHAT. A straight road through the origin with slope $m$ is $y = m x$ ( $m$ tells you how steep: $m = 0$ is flat, $m = 1$ is the $4 5^{\circ}$ diagonal). Substitute:

=\frac{\overbrace{m\,x^2}^{\text{top}}}{\underbrace{x^2(1+m^2)}_{\text{bottom}}} =\frac{m}{1+m^2}.$$ Read the cancellation term-by-term: the top is $x\cdot mx = m x^2$; the bottom factors as $x^2(1+m^2)$; the shared $x^2$ **cancels** (legal because on the road, away from the origin, $x\ne 0$). Every $x$ vanishes — the height on that road is the **constant** $\dfrac{m}{1+m^2}$, the same at every distance. **WHY this is the death blow.** The result has **no $x$ left in it** — only $m$. So the limiting value is *decided entirely by the direction* $m$. Different roads, different constants: | slope $m$ | value $\dfrac{m}{1+m^2}$ | |---|---| | $0$ (axis) | $0$ | | $1$ (diagonal) | $\tfrac12$ | | $-1$ | $-\tfrac12$ | The picture stacks several roads coloured by their constant height — a fan of different shades all meeting at the origin. ![[deepdives/dd-maths-4.4.02-d2-s04.png]] **Verdict.** The flat road says $0$, the diagonal says $\tfrac12$. Two paths, two answers ⟹ **the limit does not exist.** No single $L$ can be near both $0$ and $\tfrac12$ inside one tiny disk. --- ## Step 5 — Every case at once with polar coordinates The line test checked one direction at a time. **Polar coordinates** check *all* directions in a single formula — this is the tool [[Polar Coordinates]] gives us, and it is the right tool because our function only cares about direction. **WHAT.** Write the point by its distance $r$ from the origin and its angle $\theta$ (theta) measured from the positive $x$-axis: $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad x^2+y^2=r^2.$$ Approaching the origin is now simply $r\to 0$, at any fixed heading $\theta$. Substitute: $$f=\frac{(r\cos\theta)(r\sin\theta)}{r^2} =\frac{r^2\cos\theta\sin\theta}{r^2} =\cos\theta\sin\theta.$$ Term-by-term: top is $r^2\cos\theta\sin\theta$, bottom is $r^2$, the $r^2$ cancels, and **the distance $r$ is completely gone.** What remains, $\cos\theta\sin\theta$, depends *only on the heading $\theta$*. **WHY this settles it forever.** If the answer still holds a $\theta$ after $r\to 0$, that surviving $\theta$ **is** the path-dependence, made algebraic. Here $\cos\theta\sin\theta$ ranges from $-\tfrac12$ (at $\theta=135^\circ$) up to $+\tfrac12$ (at $\theta=45^\circ$) and is $0$ on the axes — exactly matching every line we tried, all quadrants covered in one shot. The figure plots $\cos\theta\sin\theta$ around the full circle: a curve that refuses to be one flat number. ![[deepdives/dd-maths-4.4.02-d2-s05.png]] > [!formula] The polar verdict rule > After substituting $x=r\cos\theta,\,y=r\sin\theta$ and letting $r\to0$: > - **No $\theta$ survives** (or $|f-L|\le g(r)$ with $g(r)\to0$ for *all* $\theta$) ⟹ limit exists. > - **$\theta$ survives** ⟹ direction matters ⟹ limit **does not exist**. --- ## Step 6 — The degenerate cases you must not skip > [!intuition] Cover every corner > A walkthrough is only honest if it names the weird inputs. Three of them: **(a) The origin itself.** At $(0,0)$ the bottom $x^2+y^2=0$, so $f$ is *undefined* there — division by zero. That is fine: a limit asks about points *near* the origin, never the origin ($0<\sqrt{x^2+y^2}$ in the definition, the strict $0<$). We never plug in $(0,0)$. **(b) The axes as edge directions.** On the axes $xy=0$ so $f=0$, and in polar $\theta\in\{0,90^\circ,180^\circ,270^\circ\}$ gives $\cos\theta\sin\theta=0$. These are the *only* directions that read $0$ — they are the special seam, not the rule. **(c) All four quadrants.** $\cos\theta\sin\theta$ is $+$ in quadrants I and III (where $x,y$ share sign) and $-$ in quadrants II and IV. The picture below tiles the plane by sign so you can see the checkerboard of highs and lows meeting at the centre — the visual signature of a doomed limit. ![[deepdives/dd-maths-4.4.02-d2-s06.png]] --- ## Step 7 — Contrast: what a **surviving** limit looks like To see the failure clearly, watch a nearby function *succeed*: $\dfrac{x^2y}{x^2+y^2}$ (the parent's Example 3). In polar, $$\frac{(r\cos\theta)^2(r\sin\theta)}{r^2} =\frac{r^3\cos^2\theta\sin\theta}{r^2} =r\cdot\cos^2\theta\sin\theta.$$ The key difference: an $r$ **survives out front.** Since $|\cos^2\theta\sin\theta|\le 1$, $$|f|\le r\xrightarrow[r\to0]{}0\quad\text{for every }\theta.$$ The $\theta$-part is trapped inside a factor that shrinks with $r$ regardless of direction — this is the [[Squeeze Theorem]] working on the *whole disk*. The figure overlays the doomed function (no $r$, flat coloured wedges) against the healthy one (an $r$ out front, whose whole surface funnels to $0$). ![[deepdives/dd-maths-4.4.02-d2-s07.png]] > [!recall]- Question yourself > Why does an $r$ out front save the limit but a lone $\theta$ dooms it? ::: The $r$ shrinks to $0$ no matter the direction, so a single shrinking wall (Squeeze) traps every path at $0$. A lone $\theta$ has no shrinking wall — different headings lock onto different constants. > Where is $\frac{xy}{x^2+y^2}$ undefined, and does that break the limit? ::: Only at $(0,0)$; it does not break the limit definition, which excludes the centre point itself. --- ## The one-picture summary ![[deepdives/dd-maths-4.4.02-d2-s08.png]] One image, the whole argument: axis roads reading $0$, the diagonal reading $\tfrac12$, the polar wheel of values $\cos\theta\sin\theta$ swinging between $-\tfrac12$ and $+\tfrac12$, and the verdict banner — **paths disagree ⟹ limit DNE.** > [!recall]- Feynman retelling — the whole walk in plain words > You are walking to a lamppost in a foggy field, and the "height of the ground" is our function. First you check: is the ground even defined *at* the post? No — right at the centre it's a bottomless hole (you'd divide by zero), but that's okay, we only care about the ground *around* it. You walk in along the flat east road: ground stays at $0$ the whole way. Walk the north road: also $0$. Tempting to declare "the ground is smooth, height $0$!" — but you only tried two roads. So you try the $45^\circ$ diagonal, and suddenly the ground sits at height $\tfrac12$ all along that road. Two friends arriving from two roads see two different heights right at the post — the ground has a *crack* there. To check every direction at once you switch to "how far, which way" ($r$ and $\theta$): the algebra spits out $\cos\theta\sin\theta$ — the "how far" $r$ cancelled clean away, leaving only "which way" $\theta$. That surviving direction-word *is* the crack. Compare a friendlier post where the algebra leaves an $r$ out front: as you close in, $r$ shrinks to zero from *every* direction, so everyone agrees on height $0$ — that limit lives. **The one lesson: if distance cancels and direction survives, the limit is dead.** > [!mnemonic] Distance cancels, Direction survives → DNE. And: *Lines Lie, the Wheel (polar) Tells All.* --- ## Connections - [[Multivariable Calculus|Limits and continuity in 2D — path-dependence issue]] — parent topic - [[Polar Coordinates]] — the $r,\theta$ substitution that checks all directions at once - [[Squeeze Theorem]] — the shrinking-$r$ wall that *proves* a limit exists - [[Epsilon-Delta Definition]] — the precise disk-based meaning of "close" - [[Continuity in 1D]] — the two-road world we are generalising from - [[Partial Derivatives]] — limits taken along axis-directions - [[Differentiability in 2D]] — needs continuity, which needs path-independence