Introduction

When you drop a pin on a destination in a map app, two kinds of information show up. One is "what is actually at that spot right now," and the other is "where does the road leading there head." Usually these point to the same place, but if the destination building alone were to vanish, the road leading to it would not vanish with it. The road still heads for that spot.

We can ask the same two questions about a function.

  • Where it lands: when \(x\) is exactly some value \(a\), how high does the function actually stamp itself? This is the function value \(f(a)\).
  • Where it heads: as we bring \(x\) closer and closer to \(a\), where does the function's height head? This is the limit value, written like this.
$$\lim_{x \to a} f(x)$$

Here \(\lim\) is short for limit and means "the value it heads toward," while \(x \to a\) means "let \(x\) approach \(a\)." The arrow denotes approach, not arrival.

The one sentence of this post is this. The limit value is not the value once you arrive at the point; it is the value you head toward as you approach the point. So the function value and the limit value are usually equal, but in principle they are two different events, and they can disagree.

1. A limit deliberately refuses to look at 'the point itself'

There is a part of the definition of a limit that is most often missed. When we say \(x\) approaches \(a\), \(x\) merely gets close to \(a\); it never reaches \(a\). In other words, a limit inspects only the neighbors on either side of \(a\), and it sets the point \(a\) itself aside, out of the calculation.

Why set it aside on purpose? Because that is exactly what separates "heading toward" completely from "arriving." If a limit also peeked at the value at \(a\), that would just be asking for the function value again, not new information. The reason a limit is worth anything is precisely that, even with \(a\) excluded, it can still tell us "where does this neighborhood gather."

  • Intuition: a limit only walks up near the destination and judges, "this road heads over there." It does not open the destination's door.
  • Explanation: \(x \to a\) does not include \(x = a\). Whether you approach from the left or from the right, you look only up to just before \(a\).
  • Equation: so when computing the limit value at \(a\), you do not substitute \(f(a)\); you watch what height the function narrows toward near \(a\).

This "set it aside on purpose" property explains everything else in this post. Touch a single point, and the limit does not flinch.

2. Why touching a single point leaves the limit unchanged

The function value \(f(a)\) is a single stamp pressed at the one point \(a\). Erase that stamp, or re-press it at the wrong height, and the neighbors on either side of \(a\) do not change in the slightest. Yet a limit judges by looking only at those neighbors, so no matter what you do to the stamp, the limit value stays the same.

Consider a concrete example. Take the following function.

$$f(x) = \frac{x^2 - 1}{x - 1}$$

Substitute \(x = 1\) directly and both numerator and denominator become \(0\), giving \(\tfrac{0}{0}\) — the "not yet known" form. The function value \(f(1)\) is undefined. But wherever \(x \ne 1\), we can factor the numerator and cancel.

$$\frac{x^2 - 1}{x - 1} = \frac{(x+1)(x-1)}{x-1} = x + 1 \quad (x \ne 1)$$

So this function is exactly the same as the line \(y = x + 1\) everywhere except at \(x = 1\). Let \(x\) approach \(1\), and the height heads straight for \(1 + 1 = 2\).

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$

The function value \(f(1)\) does not exist, yet the limit value stands perfectly well at \(2\). In the graph, the line \(y = x + 1\) has a single hole punched out right at \((1, 2)\). The road heads for \((1, 2)\), but there is no building at that spot.

Let us push one step further. What if someone stamps a brand-new point at the hole, at the wrong height, say \((1, 1)\)? The function value is now defined as \(f(1) = 1\). Even so, the neighbors on both sides are still \(y = x + 1\), so the value being approached remains \(2\). In this case the function value is \(1\) and the limit value is \(2\); the two openly disagree — and all we did was touch one point.

3. The three scenes where they disagree

We can now organize the relationship between heading-toward and landing-on into three scenes. In the interactive below, switch modes and check for yourself. The height headed toward as you approach from the left and from the right (open circle) and the function value actually stamped at \(x = 1\) (filled dot) are marked separately.

Where it heads vs where it lands
Drag the point x toward 1 from the left and from the right, and the height the function heads toward (the limit value) narrows in along the arrows. In smooth mode that value coincides exactly with the function value stamped at x=1; in hole mode the function value alone is moved down and disagrees; in jump mode the left and right head for different heights, so the value being approached is not pinned to a single number at all.

Scene 1 — Smooth (limit value = function value). The line \(y = x + 1\) left untouched. Let \(x\) approach \(1\) and the height heads for \(2\); the function value stamped at \(x = 1\) is also \(2\). The value approached and the value landed on coincide exactly. This very coincidence is what mathematics calls continuity.

$$\lim_{x \to 1} f(x) = f(1) = 2$$

Scene 2 — Hole (the limit value exists but differs from the function value). The case from Section 2. Both sides still head for \(2\), but \((1, 2)\) is punched out and the function value is stamped elsewhere, say at \((1, 1)\). The approached value \(2\) and the landed value \(1\) disagree.

$$\lim_{x \to 1} f(x) = 2, \qquad f(1) = 1 \quad \Rightarrow \quad \text{they differ}$$

Such a disagreement can be erased by fixing just that one point, so it is called a removable discontinuity (a hole). Re-stamp \((1, 2)\) at the hole and it becomes smooth at once.

Scene 3 — Jump (the limit value itself does not exist). This time the left and right head for altogether different heights. Suppose for \(x < 1\) the function follows \(y = x + 1\) and heads for \(2\), while for \(x \ge 1\) it follows \(y = x\) and heads for \(1\). The value approached from the left (the left-hand limit) is \(2\), and the value approached from the right (the right-hand limit) is \(1\).

$$\lim_{x \to 1^-} f(x) = 2, \qquad \lim_{x \to 1^+} f(x) = 1$$

Here \(1^-\) means "approaching from a little below \(1\)" and \(1^+\) means "approaching from a little above \(1\)." When the left and the right head for different places, "the value being approached" is not pinned to a single number. In that case we say the limit value does not exist. The two sides never reached agreement, so the very notion of heading-toward fails to hold.

4. So what exactly is 'continuity'?

Summed up in one line, the relationship between the function value and the limit value comes in exactly three flavors.

  • They are equal → smooth (continuous).
  • The limit value exists but differs from the function value, or the function value is missing → hole (removable discontinuity).
  • Left and right differ, so the limit value does not exist at all → jump (non-removable discontinuity).

From this classification the definition of continuity follows on its own. To say a function is continuous at \(x = a\) means that the value approached and the value landed on coincide exactly.

$$\lim_{x \to a} f(x) = f(a)$$

This one-line equation in fact demands three things at once. First, the \(f(a)\) on the right must exist (a stamp must be pressed); second, the limit value on the left must exist (both sides must head for the same place); and third, those two must be the same value. Break any one of the three and the function is discontinuous there. The familiar phrase "drawn without lifting the pen" is just a picture of these three conditions being met at every point.

Closing

The most confusing spot when first learning limits is the thought, "can't I just substitute \(a\) for \(x\)?" Most of the time that does give the right answer, because when a function is continuous the value approached and the value landed on are the same. But the key is to understand that the reason it always works is precisely continuity, and that the moment continuity breaks, substitution gives the wrong answer.

This single distinction — that a limit asks about heading-toward, not arriving — is the starting point for why both the derivative (the slope of an instant) and the integral (the accumulation of area) that you will meet later are written in the language of "heading toward." The instantaneous change that the landed value alone cannot explain is caught, instead, by the value being approached.