The Limit of
a Function

A value the formula can't give you

Closing in on $x = 2$

By the end of this section you will be able to

Learning objectives

1. Describe the limit of a function in correct notation definition
2. Use a table of values to estimate a limit — or see it fail Table 2.2.2
3. Use a graph to estimate a limit — or see it fail Figure 2.2.4
4. Define one-sided limits and give examples Def 2.2.2
5. Relate one-sided and two-sided limits Thm 2.2.2
6. Describe an infinite limit in correct notation Def 2.2.3
7. Define a vertical asymptote Def 2.2.4

Why we approach instead of substitute

Why the hole?

Definition 2.2.1 — Limit (intuitive)

Definition 2.2.1 — Limit

Let $f(x)$ be defined for all $x$ in an open interval containing $a$, except possibly at $a$ 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 2.2.2: The limit reports the neighborhood (shaded band around $a$), not the value at the point.

Read it: for $f(x)=\dfrac{x^2-4}{x-2}$, the value $f(2)$ does not exist — yet $\displaystyle\lim_{x \to 2} f(x) = 4$.

Example 2.2.1 — A limit from a table

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

Table 2.2.2: Both sides march toward 1.

Estimate

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

Figure 2.2.3: $y=\dfrac{\sin x}{x}$ settles on height 1 at $x=0$.

Try It Now 2.2.1 — Your turn

Estimate $\displaystyle\lim_{x \to 1}\frac{\tfrac{1}{x}-1}{x-1}$

Example 2.2.3 — A limit from a graph

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

Read the neighborhood

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

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

Figure 2.2.4: $g(-1)=4$, yet $\displaystyle\lim_{x\to-1}g(x)=3$.

Theorem 2.2.1 — Two important limits

From estimates to building blocks

Tables and graphs rest on guesswork — a trend read, not a value proved. §2.3 builds algebraic limit laws that compute limits exactly. Two limits are so basic that every one of those laws leans on them.

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

Two ways to fail · Example 2.2.4

A two-sided limit exists only if $f(x)$ approaches a single real number from both sides. It can miss in two ways: oscillation (bounces forever) or blow-up (shoots to $\pm\infty$). Both read DNE for now.

Evaluate $\displaystyle\lim_{x \to 0}\sin\!\left(\tfrac{1}{x}\right)$. As $x \to 0$, $1/x$ blows up, so $\sin(1/x)$ oscillates faster and faster — never settling.

Oscillation → DNE

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

Figure 2.2.5: $\sin(1/x)$ oscillates without bound as $x\to 0$.

Some failures deserve their own notation

Definition 2.2.2 — One-sided limits

One-sided limits

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

Example 2.2.5 — Evaluating one-sided limits

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

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

Sides disagree → DNE

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

Figure 2.2.7: A jump at $x=2$: left limit 3, right limit 0.

Try It Now 2.2.4 — Your turn

One-sided limits of $\dfrac{|x^2-4|}{x-2}$ at $x=2$

Theorem 2.2.2 — Relating one- and two-sided limits

When the two sides shake hands

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

Recording how a limit fails

Definitions 2.2.3 & 2.2.4

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

Example 2.2.6 — Recognizing an infinite limit

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

Table 2.2.4: The two sides blow up in opposite directions.

Opposite infinities → DNE

$$\lim_{x \to 0^-}\frac{1}{x}=-\infty,\qquad \lim_{x \to 0^+}\frac{1}{x}=+\infty.$$So $\displaystyle\lim_{x \to 0}\frac{1}{x}$ DNE.

Figure 2.2.9: $y=\dfrac{1}{x}$ with a vertical asymptote at $x=0$.

Theorem 2.2.3 — Infinite limits from positive integers

The parity of $n$ sets the sign

For a real number $a$ and a positive integer $n$, either way the line $x = a$ is a vertical asymptote.

Example 2.2.7 & Try It Now 2.2.6

Example 2.2.8 — Reading a graph

$x=-4$ (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 2.2.10: One graph, four behaviors: smooth, hole, jump, asymptote.

Four behaviors, one verdict each

Where is a function headed near a point — even when the point 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

Table 2.2.5: The limit always reads the neighborhood.

When the neighborhood agrees, the limit exists; when it splits or blows up, we don't stop at "DNE" — we name how.

Figure 2.2.10: The four verdicts, all on one graph.