3.2 Limits Along Different Roads

In one variable, a point can be approached from only two sides, and a limit exists when left and right agree. In the plane there are infinitely many roads into any point — every direction, every curve — and ${lim}_{(x, y) \to (a, b)} f (x, y)$ exists only if all of them agree. That is a far stronger demand, and this section builds a surface on which you can watch it fail: different straight roads into the origin genuinely arrive at different heights.

The specimen is

$f (x, y) = \frac{2 x y}{x^{2} + y^{2}},$

defined everywhere except the origin. Away from $open parenthesis 0, 0 close parenthesis$ its graph is smooth ridges and valleys, but at the origin it pleats into a sharp vertical fin — and the fin, it will turn out, is the failed limit.

A red-and-blue surface that pleats into a sharp vertical fin at the origin, ridges at height 1 along the diagonal y = x and valleys at height -1 along y = -x. Three thick straight roads ride the surface through the origin: a green one flat at height 0, a red one at height 1, and a blue one at height -1. A fourth amber road is hidden. — The red-and-blue surface $z = 2 x y / (x^{2} + y^{2})$ (the expression carries a tiny $plus 0.000001$ in the denominator so the mesh stays finite at the origin), pleating into a vertical fin at $open parenthesis 0, 0 close parenthesis$ . Three thick straight roads ride the surface through the origin: green along $y equals 0$ at height $0$ , red along $y = x$ at height $1$ , and blue along $y = - x$ at height $minus 1$ . A fourth, amber road along $y equals 2 x$ hides in the object list.

Explore

Orbit slowly around the origin. Each colored road lies in a vertical plane over a straight line in the $x y$ -plane. For each of the three visible roads, read off two things: the line it travels over, and the height it travels at.
Every road is perfectly level — constant height — yet all three pass over the same point $open parenthesis 0, 0 close parenthesis$ . What are the three heights over the origin? Could any single value $f open parenthesis 0, 0 close parenthesis$ agree with all three roads at once?
The hidden amber road runs along $y equals 2 x$ , a line that sits between $y equals 0$ (green, height $0$ ) and $y = x$ (red, height $1$ ). Before revealing it, predict: will its height land between $0$ and $1$ ? Closer to which? Commit to a number.
Reveal the amber road with its eye toggle in the object list, orbit to a side view, and read its height against the grid. How close was your guess? The example below does the algebra.
Now make the scene grade your computation. Edit the amber road's $y$ -component from 2*t to 5*t: the road jumps off the surface, because its $z$ -component still carries the old height. Compute the correct height for slope $5$ , retype the $z$ -component, and watch — if the road settles back onto the surface, your value is right.
All your computations follow one formula: the road over $y = k x$ rides at height $2 k / open parenthesis 1 plus k squared close parenthesis$ . Which slope rides highest? Which lowest? Sketch height against $k$ on paper.

Why the limit fails

Along any line through the origin the function is constant, so each road is level — but the constants differ from road to road, filling the whole range from $minus 1$ to $1$ . For the limit to exist, one number would have to be the destination of every road at once. No number is, so

${lim}_{(x, y) \to (0, 0)} \frac{2 x y}{x^{2} + y^{2}} does not exist.$

Continuity at the origin is then beyond rescue: it would require $f open parenthesis 0, 0 close parenthesis$ to equal a limit that is not there. No choice of value plugs the fin — the fin is the disagreement between roads, made solid.

Example 1 Every straight road has its own altitude

Find the height at which the surface carries the road along $y = k x$ , then verify the amber road's height and compute the correct height after its slope is edited to $5$ .

Substitute $y = k x$ with $x not equal to 0$ :

$f (x, k x) = \frac{2 x (k x)}{x^{2} + k^{2} x^{2}} = \frac{2 k}{1 + k^{2}},$

a constant — the same for every $x not equal to 0$ , which is why each lifted road is level. For $k equals 2$ the height is $\frac{4}{5} = 0.8$ , matching the amber road. For $k equals 5$ it is $\frac{10}{26} = \frac{5}{13} \approx 0.385$ — the value that reseats the edited road. The extremes come from $open parenthesis 1 ∓ k close parenthesis squared greater than or equal to 0$ , which rearranges to $| 2 k | less than or equal to 1 plus k squared$ : the height $2 k / open parenthesis 1 plus k squared close parenthesis$ never leaves $[ minus 1, 1 ]$ , and touches $± 1$ exactly at $k equals ± 1$ — the red and blue roads. Different roads, different constants: the two-variable limit does not exist.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Aligned to OpenStax Calculus Volume 3 (Strang & Herman), © OpenStax (Rice University), licensed CC BY-NC-SA 4.0; no OpenStax content is reproduced, and this work is not affiliated with or endorsed by OpenStax or Rice University. License: CC-BY-NC-SA-4.0.