AP Calculus AB

From average change to change at an instant

The average rate of change of a function $f$ over an interval from $x = a$ to $x = b$ is

the slope of the secant line through the two points on the graph. It divides the change in $f$ by the change in $x$, so it is undefined when the two values of $x$ coincide, because the denominator would be zero. That is the difficulty with asking how fast something is changing at a single instant: there is no interval, and the quotient becomes $\frac{0}{0}$.

The way round it is to shrink the interval. If $f(t)$ is the distance a car has travelled at time $t$, the average speed over $t = 2$ to $t = 2.1$, then over $t = 2$ to $t = 2.01$, then $t = 2$ to $t = 2.001$, forms a list of numbers that settles towards one value. That value is the speed at the instant $t = 2$. A limit is the tool that makes this precise: it describes the value a quantity is approaching, without requiring that the quantity ever reaches it.

Definition and notation

Given a function $f$ and a number $c$, the limit of $f(x)$ as $x$ approaches $c$ is a real number $L$ if $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$, but not equal to $c$. It is written

The words "but not equal to $c$" matter. The limit records what happens near $c$ and says nothing on its own about what happens at $c$. The function might be undefined at $c$, might have a value that differs from $L$, or might have the value $L$, and the limit is the same in all three cases. So $\lim_{x \to c} f(x)$ and $f(c)$ are different things, and keeping them apart is what the whole idea of continuity later rests on.

One-sided limits

Sometimes the values approached from the left of $c$ differ from those approached from the right. The left-hand limit $\lim_{x \to c^-} f(x)$ uses only values with $x < c$, and the right-hand limit $\lim_{x \to c^+} f(x)$ uses only values with $x > c$. The two-sided limit exists, and equals $L$, exactly when both one-sided limits exist and both equal $L$:

A limit at an endpoint of a domain, such as $\lim_{x \to 0^+} \sqrt{x}$, is necessarily one-sided, because the function does not exist on the other side.

Estimating limits from a graph

To read $\lim_{x \to c} f(x)$ from a graph, follow the curve towards $x = c$ from the left and then from the right, and note the height it heads for each time. The height of the point actually plotted at $x = c$ plays no part.

Three situations come up repeatedly.

• A curve with an open circle at $(c, L)$ and a filled dot at $(c, M)$ with $M \neq L$: the limit is $L$, and $f(c) = M$.
• A curve with a hole at $(c, L)$ and nothing at $x = c$: the limit is $L$, and $f(c)$ does not exist.
• A curve that jumps, arriving at height $p$ from the left and leaving from height $q \neq p$ on the right: the one-sided limits are $p$ and $q$, and the two-sided limit does not exist.

A graph drawn by a calculator or computer has a limited window and a limited resolution, so it can hide a short vertical jump or a very narrow spike near the point of interest. Zooming in and checking the function's formula are both safeguards against misreading a limit from a graph.

Estimating limits from a table

A table gives values of $f$ at chosen inputs. To estimate $\lim_{x \to c} f(x)$ from one, look at inputs on both sides of $c$ that get progressively closer to it, and see whether the outputs head towards a single number.

For $f(x) = \dfrac{x^2 - 9}{x - 3}$, which is undefined at $x = 3$:

Both sides approach $6$, so the estimate is $\lim_{x \to 3} f(x) = 6$. The estimate is backed by algebra, since $\frac{x^2 - 9}{x - 3} = x + 3$ for $x \neq 3$, but a table by itself is evidence rather than proof. A finite list of values cannot rule out behaviour that happens closer to $c$ than the smallest gap tabulated, and a table with values on only one side of $c$ cannot show that the two-sided limit exists.

When a limit does not exist

A limit fails to exist, as a real number, in three main ways.

1. The one-sided limits disagree. For $f(x) = \dfrac{|x|}{x}$, the function equals $-1$ for $x < 0$ and $1$ for $x > 0$, so $\lim_{x \to 0^-} f(x) = -1$ and $\lim_{x \to 0^+} f(x) = 1$, and $\lim_{x \to 0} f(x)$ does not exist.
2. The function is unbounded near $c$. For $f(x) = \dfrac{1}{x}$ as $x \to 0$, the values grow without bound in size, positive from the right and negative from the left. Even when both sides grow in the same direction, as for $\dfrac{1}{x^2}$, there is no real-number limit. Such behaviour is described with infinite limits and vertical asymptotes.
3. The function oscillates without settling. For $f(x) = \sin\!\left(\dfrac{1}{x}\right)$ as $x \to 0$, the function swings between $-1$ and $1$ infinitely often in any interval around $0$, so it does not approach any one value.

Worked example

Let

Find $\lim_{x \to 2} f(x)$.

Approaching from the left uses the first rule, so $\lim_{x \to 2^-} f(x) = 2(2) + 1 = 5$. Approaching from the right uses the second rule, so $\lim_{x \to 2^+} f(x) = 7 - 2 = 5$. The one-sided limits agree, so $\lim_{x \to 2} f(x) = 5$, and here this also equals $f(2) = 5$.

Changing the second rule to $8 - x$ gives a right-hand limit of $6$ and a left-hand limit of $5$, so the two-sided limit would not exist.

Interpreting limit statements

A limit statement can be read in words, as a graph feature or as a number in a table, and moving between these is part of the skill. $\lim_{x \to 4} f(x) = 9$ says that $f(x)$ gets as close to $9$ as one likes when $x$ is taken close enough to $4$ on either side, that the graph near $x = 4$ heads for the height $9$, and that table entries for $x$ near $4$ crowd round $9$. It says nothing about $f(4)$, which may be $9$, may be another number, or may not exist.