Intuitive Definition of Limit, calculus lesson

Try to evaluate the function

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

at $x = 1$ and you get $0/0$ , which is not a number. Yet for $x$ close to $1$ the outputs sit very close to $2$ , and nothing strange happens nearby. A limit is the tool that captures this. It asks where the outputs are heading, and it does not care what happens at the point itself.

Key terms

Approaching a value

Definition Let $f$ be defined on an open interval containing $a$ , except possibly at $a$ itself. We say that $f(x)$ approaches the number $L$ as $x$ approaches $a$ if the values $f(x)$ can be made as close to $L$ as we like by taking $x$ sufficiently close to $a$ , on either side of $a$ , but not equal to $a$ .

In plain words However small a tolerance around $L$ you ask for, every input close enough to $a$ (other than $a$ ) gives an output within that tolerance.

Example Here are some outputs of the function $f$ from the opening for inputs near $a = 1$ .

$x$	$0.9$	$0.99$	$0.999$	$1.001$	$1.01$	$1.1$
$f(x)$	$1.9$	$1.99$	$1.999$	$2.001$	$2.01$	$2.1$

From both sides the values close in on $2$ . Algebra explains why. For every $x \neq 1$

\frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1,

and $x + 1$ is near $2$ when $x$ is near $1$ .

Nearby inputs

Definition The limit of $f$ at $a$ depends only on the values $f(x)$ for $x$ in some open interval around $a$ with $x \neq a$ . In particular, if $f(x) = g(x)$ for every $x \neq a$ in some open interval containing $a$ , then $f$ and $g$ have the same limit at $a$ , or neither has one.

In plain words Only the neighborhood counts. What $f$ does far away from $a$ , and what it does exactly at $a$ , has no effect on the limit.

Example The function $f$ from the opening and the function $g(x) = x + 1$ agree at every $x \neq 1$ . The first is undefined at $1$ and the second equals $2$ there, but both approach $2$ as $x$ approaches $1$ . Changing $g$ at $x = 50$ to any value you like would not change that either.

Limit notation

Definition When $f(x)$ approaches $L$ as $x$ approaches $a$ , we write

\lim_{x \to a} f(x) = L,

read “the limit of $f(x)$ as $x$ approaches $a$ is $L$ ”. An equivalent notation is $f(x) \to L$ as $x \to a$ . Here $L$ must be a real number.

In plain words The symbol $\lim_{x \to a}$ is an instruction. Look at $f(x)$ for $x$ near $a$ and report the number the outputs settle on.

Example For $f(x) = 2x + 1$ , inputs near $3$ give outputs near $7$ , since

\lvert (2x + 1) - 7 \rvert = 2\lvert x - 3 \rvert

is small whenever $\lvert x - 3 \rvert$ is small. So $\lim_{x \to 3} (2x + 1) = 7$ .

Function value versus limiting value

Definition The value $f(a)$ and the limit $\lim_{x \to a} f(x)$ are different objects. The limit can exist while $f(a)$ is undefined, the two can both exist and differ, and $f(a)$ can exist while the limit does not.

In plain words The value $f(a)$ is where the function actually is at $a$ . The limit is where it is heading as you come close to $a$ . They often match, but nothing forces them to.

Example Let

g(x) = \begin{cases} x^2, & x \neq 2, \\ 1, & x = 2. \end{cases}

Then $g(2) = 1$ , while for $x$ near $2$ but not equal to it $g(x) = x^2$ is near $4$ . So $\lim_{x \to 2} g(x) = 4 \neq g(2)$ . On a graph this looks like the parabola with one point lifted out of it and dropped to height $1$ .

Graphs and numerical tables

Definition A table of values $f(x)$ for inputs $x$ approaching $a$ from both sides, or a graph of $f$ near $a$ , gives evidence for a limit and a guess for its value. Neither is a proof, because a table checks only finitely many inputs and a graph shows only what its scale and resolution allow.

In plain words Tables and graphs are good for guessing. Confirm the guess with algebra or with the precise definition that comes later in this unit.

Example A table supports a true guess for $(\sin x)/x$ near $0$ , with $x$ in radians.

$x$	$0.5$	$0.1$	$0.01$	$0.001$
$\frac{\sin x}{x}$	$0.958851$	$0.998334$	$0.999983$	$0.9999998$

The function is even, so negative inputs give the same values, and the guess

\lim_{x \to 0} \frac{\sin x}{x} = 1

is correct. Now take $h(x) = \sin (\pi/x)$ and the inputs $x = 1, 0.1, 0.01, 0.001$ . Every one of them gives $h(x) = \sin(n\pi) = 0$ for a whole number $n$ , so the table suggests a limit of $0$ at $x = 0$ . That guess is wrong, as the next term shows.

Oscillation

Definition A function oscillates near $a$ when, in every open interval around $a$ , its values keep swinging between numbers that stay a fixed distance apart. If there is a fixed $d > 0$ such that every open interval around $a$ contains inputs $x_1, x_2 \neq a$ with $\lvert f(x_1) - f(x_2) \rvert \ge d$ , then $\lim_{x \to a} f(x)$ does not exist.

In plain words The outputs never settle down. No matter how close to $a$ you zoom in, they still jump back and forth by at least $d$ .

Example Take $f(x) = \sin (1/x)$ near $0$ . For every positive whole number $k$ the inputs

x_k = \frac{1}{\pi/2 + 2k\pi} \quad \text{and} \quad y_k = \frac{1}{3\pi/2 + 2k\pi}

give $f(x_k) = 1$ and $f(y_k) = -1$ . Both kinds of input get as close to $0$ as we like as $k$ grows, so every interval around $0$ contains outputs $1$ and $-1$ . With $d = 2$ the limit fails to exist. The same reasoning applies to $\sin (\pi/x)$ , which is why the table above misled us.

Existence and nonexistence

Definition The limit $\lim_{x \to a} f(x)$ exists when there is a real number $L$ that $f(x)$ approaches as $x$ approaches $a$ from both sides. Otherwise the limit does not exist. The common ways to fail are a jump, where the two sides head to different numbers, unbounded growth, where the values do not stay near any real number, and oscillation.

In plain words A limit exists only when one single real number works from both sides. If the two sides disagree, or the values blow up, or they keep swinging, there is no limit.

Example For the step function $H(t) = 0$ when $t < 0$ and $H(t) = 1$ when $t \ge 0$ , inputs just left of $0$ give $0$ and inputs just right of $0$ give $1$ , so $\lim_{t \to 0} H(t)$ does not exist. For $1/x^2$ near $0$ , the inputs $x = \pm 0.01$ already give $10000$ , and the outputs grow past every bound, so there is no real limit at $0$ either.

Check yourself

Let $f(x) = x + 3$ for $x \neq 1$ and $f(1) = 0$ . Find $f(1)$ and $\lim_{x \to 1} f(x)$ .

Show the answer

The value is $f(1) = 0$ , but for $x$ near $1$ and $x \neq 1$ the values $x + 3$ are near $4$ , so $\lim_{x \to 1} f(x) = 4$ .

Guess the limit below and then confirm it with algebra.

\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}

Show the answer

Since $x - 4 = (\sqrt{x} - 2)(\sqrt{x} + 2)$ for $x \ge 0$ , the quotient equals $\sqrt{x} + 2$ for $x \ge 0$ with $x \neq 4$ . That approaches $\sqrt{4} + 2 = 4$ , so the limit is $4$ .

Does $\lim_{x \to 0} \cos (1/x)$ exist?

Show the answer

No. For every positive whole number $k$ the function equals $1$ at $x = 1/(2k\pi)$ and $-1$ at $x = 1/((2k + 1)\pi)$ , and both kinds of input come arbitrarily close to $0$ . The function oscillates, so there is no limit.

Intuitive Definition of Limit

Theory, part 3

Practice, part 4

Key terms

Approaching a value

Nearby inputs

Limit notation

Function value versus limiting value

Graphs and numerical tables

Oscillation

Existence and nonexistence

Check yourself

How this lesson connects

Used later in