A tricky limit

Here’s just a little discussion of limits including notation, meaning, and a numerical example.

What even is a limit?

Here’s a little bit of slightly heavy notation: \[\lim_{x\to a}f(x) = L.\]

We read this as:

The limit as \(x\) goes to \(a\) of \(f(x)\) equals \(L\).

But, what do we mean?

\(\lim_{x\to a}f(x) = L\) means that we can make the value of \(f(x)\) as close as we like to \(L\) by taking the value of \(x\) as close as necessary to \(a\), but not equal to \(a\).

What could be easier?

Example

Here’s the graph of

\[f(x)=\frac{2^x - 1}{x}:\]

viewof trace = Inputs.toggle({ label: "Trace point:", value: false })

viewof x = {
  let range = Inputs.range([-2.5, 2.5], {
    value: 1,
    step: 0.01,
    disabled: !trace
  });
  d3.select(range).select("input").remove();
  return range;
}

graph = {
  let pts = d3.range(-3, 3, 0.011).map((x) => [x, f(x)]);
  let w = 800;
  let h = 500;
  let marks = [
    Plot.axisY({ x: 0 }),
    Plot.ruleX([0]),
    Plot.line(pts),
    Plot.dot([[0, Math.log(2)]], { r: 4, fill: "white", stroke: "black" }),
    Plot.ruleY([0]),
    Plot.axisX({ y: 0 })
  ];
  if (trace) {
    marks.push(Plot.dot([[x, f(x)]], { r: 5, fill: "black" }));
    marks.push(Plot.text([{t: `f(0)≈f(${x})=${d3.format('0.4f')(f(x))}`, x: -3, y: 2}], {x: 'x', y: 'y', text: 't', fontSize: 18, textAnchor: 'start'}))
  }
  return Plot.plot({
    marks: marks
  });
}

F = (x) => 2 ** x
f = (x) => (F(x) - 1) / x

Note that, if we plug \(x=0\) into \(f\), we get the dreaded expressions \(0/0\), which is nonsensical. In math speak, \(f\) is not defined at \(x=0\) or, equivalently, zero is not in the domain of \(f\). This is indicated by the hole we see in the graph. Note that you probably won’t see this behavior in most computer generated graphs (like this Desmos graph), unless you add it explicitly.

While we can’t compute \(f(0)\), we might hope to estimate the limit as \(x\to0\) of \(f(x)\), i.e. we can ask \[\lim\limits_{x\to0}f(x) = ?\]

That’s part of the point behind the “Trace point” toggle and slider. To be more precise, we might systematically plug in points that get closer and closer to zero and examine the values coming out That’s part of the point behind the following table:

html`<table>
  <tr style="border-bottom: solid 1px black !important">
    <td style="border-right: solid 0.6px black">${tex`x`.outerHTML}</td>${d3
  .range(7)
  .map(
    (n) =>
      `<td style="border-left: solid 1px black !important">${10 ** -n}</td>`
  )
  .join("")}
  </tr>
  <tr>
    <td style="border-right: solid 0.6px black">${tex`f(x)`.outerHTML}</td>${d3
  .range(7)
  .map(
    (n) =>
      `<td style="border-left: solid 1px black">${d3.format("0.7f")(
        f(10 ** -n)
      )}</td>`
  )
  .join("")}
  </tr>
</table>`

I guess this suggests that \[\lim\limits_{x\to0}f(x) \approx 0.69314.\] Note that I did not include digits that don’t appear to have settled down just yet.

I do wonder, why might we go through all this??