The Limit

A limit is simply the value a function is heading toward as its input gets closer and closer to some number. It's the tool that lets us talk about "the slope at a point" and, later, areas under curves, so the rest of calculus runs on it. All we're really saying is "as $x$ gets close to $a$, $f(x)$ gets close to $L$." The notation $\lim_{x\to a} f(x) = L$ is just that sentence in shorthand: $\lim$ (where is it heading?), $x\to a$ (as $x$ creeps up on $a$, from both sides), $f(x)$ (the outputs we're watching) and $L$ (the number they settle on). Notice what it doesn't say: nothing about what happens when $x$ actually equals $a$. That's the whole idea.

In the last lesson, the secant slopes $\dfrac{x^2 - 1}{x - 1}$ closed in on $2$ as $x$ slid toward $1$. We couldn't plug in $x = 1$, but we could still say where things were heading. That "where it's heading" is what a limit is, and we write

Estimating a limit. The simplest way is to do what we did with the table: plug in values of $x$ closer and closer to $a$, from both sides, and see where the outputs head. A graph tells you the same thing if you follow the curve toward $x = a$.

The value at $a$ doesn't matter. This is the part that feels strange at first. $f(x) = \dfrac{x^2 - 1}{x - 1}$ isn't even defined at $x = 1$. But for every other $x$, you can cancel and get $x + 1$, so near $1$ the outputs are near $2$, and $\lim_{x\to1} f(x) = 2$. The graph is the line $y = x + 1$ with a hole punched in it, and the limit tells you where the hole is. It makes sense that we'd want it this way: the tangent slope was defined by exactly this kind of $\frac00$ expression, so a limit that cared about the value at $a$ would be useless to us.

When limits don't exist. A limit needs the outputs to settle on one number. There are three typical ways that fails:

• Jumps: the function heads to one value from the left and a different one from the right.
• Blow-ups: the values grow without bound, like $\frac{1}{x^2}$ near 0.
• Oscillation: $\sin\frac1x$ swings between $-1$ and $1$ infinitely often near 0 and never settles down.

A common mistake. It's tempting to trust a table completely, since that's how we've been estimating. But tables can fool you. Plug $x = 0.1, 0.01, 0.001$ into $\sin\frac{\pi}{x}$ and you get $0$ every time, which looks like a limit of $0$. It isn't: those inputs happen to land exactly where the sine is zero, and in between it keeps swinging between $-1$ and $1$, so the limit doesn't exist. A table can suggest an answer. Algebra and theorems are what actually decide it.

A computer finds a derivative numerically by shrinking $h$ in $\frac{f(x+h) - f(x)}{h}$ and watching the values settle: a limit computed by table.