The Limit of a Function

§2.2 — Name the method · handout

Do it again — for any function

In Closing In you found the single value $f(x)$ was heading toward as $x$ got closer and closer to $2$ — without ever using $x = 2$ itself. The point of math is to do it again, for any function. So write down exactly what your group did, step by step — then name and define your method.

§2.2 — Name the method · define it

Write it down, then name it

You found where $f$ was heading as $x \to 2$ — without ever using $x = 2$. Write down exactly what your group did:

1. Choose $x$-values closing in on the point from the left.
2. And matching values from the right.
3. Compute $f(x)$ at each one.
4. Read down both columns — when they close in on a single number, that's where $f$ is heading.

The Closing-In Method — to find the value a function is heading toward as $x$ gets close to a point, plug in $x$-values closer and closer from both sides and watch what number the outputs settle on.

§2.2 — The limit, intuitively

Definition 2.2.1

Definition 2.2.1 — Limit (Intuitive)

Let $f(x)$ be defined at all values in an open interval containing $a$, with the possible exception of $a$ itself, and let $L$ be a real number. If all values of $f(x)$ approach $L$ as the values of $x\ (\ne a)$ approach $a$, then the limit of $f(x)$ as $x$ approaches $a$ is $L$:

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

Figure: the shaded band — the neighborhood around $a$ — is what the limit reports.

The limit reports what the neighborhood is doing, not the value at the point. For $f(x)=\dfrac{x^2-4}{x-2}$, the value $f(2)$ does not exist — yet $\displaystyle\lim_{x \to 2} f(x) = 4$.

§2.2 — Why the hole?

Why the hole — and the big idea

Factor and cancel:

So $f$ is the line $y = x + 2$ with the single point $(2, 4)$ punched out. The cancellation is only legal for $x \ne 2$ — which is why we approach the point instead of substituting it. (Evaluating limits exactly by factoring is the work of §2.3.)

Big Idea. A limit reads where a function is headed, not where it is — like reading which way a friend is heading through a crowd, even when you can't see exactly where they stand.

§2.2 — Estimate a limit from a table

Example 2.2.1

Evaluate $\displaystyle\lim_{x \to 0}\frac{\sin x}{x}$ using a table of values.

The function is even, so the left and right columns match:

$x$	$\sin x / x$
$\pm 0.1$	0.99833
$\pm 0.01$	0.99998
$\pm 0.001$	0.9999998

Both sides march toward 1: $\displaystyle\lim_{x \to 0}\frac{\sin x}{x}=1.$

Figure 2.13: $\sin x / x$ heads to 1 from both sides of 0.

§2.2 — Try it now

Try It Now 2.2.1

Estimate $\displaystyle\lim_{x \to 1}\frac{\frac{1}{x}-1}{x-1}$ using a table of values.

§2.2 — Estimate a limit from a graph

Example 2.2.3

For the function $g(x)$ shown, evaluate $\displaystyle\lim_{x \to -1} g(x)$.

The marked point shows $g(-1)=4$, but the curve approaches height 3 from both sides: $\displaystyle\lim_{x \to -1} g(x)=3.$

The limit and the function value are allowed to disagree — the limit reports the neighborhood, not the isolated point.

Figure 2.15: $g(-1)=4$ marked, but the curve heads to 3.

§2.2 — Two atomic limits

Theorem 2.2.1

The identity function $f(x)=x$ marches along with $x$; a constant function never moves. These two limits are the atomic building blocks inside every limit law we prove from here on.

§2.2 — A limit that fails: oscillation

Example 2.2.4

Evaluate $\displaystyle\lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)$.

As $x \to 0$, the input $1/x$ blows up, so $\sin(1/x)$ oscillates faster and faster — never settling on a value.

Along $x=\dfrac{2}{\pi},\dfrac{2}{3\pi},\dfrac{2}{5\pi},\dots$ the value cycles $1,-1,1,-1,\dots$ forever. Different paths to 0 give different values, so $\displaystyle\lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)$ DNE.

Figure 2.17: $\sin(1/x)$ oscillates without bound as $x \to 0$.

§2.2 — When "DNE" hides the real story

Each side can still be clean

§2.2 — One-sided limits

Definition 2.2.2

For $g(x)=\dfrac{|x-2|}{x-2}$: the two-sided limit DNE, but the side limits are clean — $\displaystyle\lim_{x \to 2^-} g(x) = -1$ and $\displaystyle\lim_{x \to 2^+} g(x) = 1$.

§2.2 — Evaluating one-sided limits

Example 2.2.5

$$f(x)=\begin{cases} x+1 & \text{if } x < 2,\\[2pt] x^2-4 & \text{if } x \ge 2.\end{cases}$$

Use the correct branch on each side: $\displaystyle\lim_{x \to 2^-} f(x) = 3,\quad \lim_{x \to 2^+} f(x) = 0.$

The side limits disagree, so $\displaystyle\lim_{x \to 2} f(x)$ DNE — the graph jumps at $x = 2$.

Figure 2.18: the branches disagree — a jump at $x=2$.

§2.2 — Try it now

Try It Now 2.2.4

Estimate the one-sided limits of $\dfrac{|x^2-4|}{x-2}$ at $x = 2$.

$\displaystyle\lim_{x \to 2^-}\frac{|x^2-4|}{x-2}=-4,\qquad \lim_{x \to 2^+}\frac{|x^2-4|}{x-2}=4.$ Answer: left side $-4$; right side $+4$ (two-sided DNE).

§2.2 — One-sided and two-sided

Theorem 2.2.2

The two-sided limit exists exactly when the two side limits shake hands on the same value. If they disagree, the two-sided limit does not exist.

§2.2 — The blow-up failure

Back to the blow-up

§2.2 — Infinite limits & vertical asymptotes

Definitions 2.2.3 & 2.2.4

Writing $+\infty$ records how the limit fails — by blowing up — instead of just labeling it "DNE". $h(x)=\dfrac{1}{(x-2)^2}$ has $\displaystyle\lim_{x \to 2} h(x) = +\infty$, so $x = 2$ is a vertical asymptote. (Infinity describes behavior, not a real-number value.)

§2.2 — Recognizing an infinite limit

Example 2.2.6

Evaluate the one- and two-sided limits of $\dfrac{1}{x}$ at $x = 0$.

$x$	$1/x$	$x$	$1/x$
-0.01	-100	0.01	100
-0.001	-1000	0.001	1000

$\displaystyle\lim_{x \to 0^-}\frac{1}{x}=-\infty,\quad \lim_{x \to 0^+}\frac{1}{x}=+\infty.$ The sides blow up in opposite directions, so $\displaystyle\lim_{x \to 0}\frac{1}{x}$ DNE.

Figure 2.19: $1/x$ blows up to $-\infty$ and $+\infty$ at $x=0$.

§2.2 — Infinite limits from positive integers

Theorem 2.2.3

Either way, the line $x = a$ is a vertical asymptote.

§2.2 — Finding a vertical asymptote

Example 2.2.7

Evaluate the limits of $f(x)=\dfrac{1}{(x+3)^4}$ at $x = -3$, and identify any vertical asymptote.

Match the template $\dfrac{1}{(x-a)^n}$: here $a = -3$ and $n = 4$ (even).

Because the limit blows up at $x = -3$, the line $x = -3$ is a vertical asymptote.

§2.2 — Try it now

Try It Now 2.2.6

Evaluate the limits of $f(x)=\dfrac{1}{(x-2)^3}$ at $x = 2$ and identify any vertical asymptote.

The sides disagree, so the two-sided limit DNE. But one side blowing up is enough. Answer: left $-\infty$; right $+\infty$; two-sided DNE; vertical asymptote at $x = 2$.

§2.2 — Four behaviors on one graph

Example 2.2.8

Read each behavior off the graph of $f$:

• $x = -4$ (smooth): $\displaystyle\lim_{x \to -4} f(x) = 0$, and $f(-4) = 0$.
• $x = -2$ (hole): $\displaystyle\lim_{x \to -2} f(x) = 3$, but $f(-2)$ is undefined.
• $x = 1$ (jump): $\lim^- = 6$, $\lim^+ = 3$, two-sided DNE, $f(1) = 6$.
• $x = 3$ (asymptote): both sides $\to -\infty$, $f(3)$ undefined.

Figure 2.21: four distinct behaviors on a single graph.

§2.2 — The question, answered

Four answers, one graph

Where is a function headed near a point — even when the point itself can't be trusted? One graph holds all four answers.

Point on the graph	Limit verdict
smooth $(x=-4)$	exists; $\lim=f(a)$
hole $(x=-2)$	exists; $\lim\ne f(a)$
jump $(x=1)$	two-sided DNE; sides clean
asymptote $(x=3)$	infinite; vertical asymptote

The limit always reads the neighborhood, never the point. When the neighborhood agrees, the limit exists; when it splits or blows up, we don't stop at "DNE" — we name how.

Figure 2.21: one graph, all four verdicts.