Calculus · Chapter 2 · Limits

The Limit of a Function

Where is a function headed near a point — even when we can't, or shouldn't, plug the point in?

bookSHelf · Calculus Volume 1 · §2.2 · a self-paced section

The Limit of a Function · bookSHelf Calculus§2.2

Learning objectives — by the end of this section you will be able to

Objectives

Using correct notation, describe the limit of a function Definition 2.2.1
Use a table of values to estimate a limit — or see it fail Examples 2.2.1, 2.2.6
Use a graph to estimate a limit — or see it fail Examples 2.2.3, 2.2.8
Define one-sided limits and give examples Definition 2.2.2
Relate one-sided and two-sided limits Theorem 2.2.2
Using correct notation, describe an infinite limit Definition 2.2.3
Define a vertical asymptote Definition 2.2.4

The Limit of a Function · bookSHelf Calculus§2.2

The opening problem — a value the formula can't give you

Closing in on $x = 2$

This is the graph of $f(x)=\dfrac{x^2-4}{x-2}$ . At $x = 2$ it has no value (you'd divide by zero). So where is the graph headed as $x$ gets close to 2? Pick $x$ -values just below and just above 2, and compute $f(x)$ .

Figure 2.2.1: $f(x)=\dfrac{x^2-4}{x-2}$ — the line $y=x+2$ with the point $(2,4)$ punched out.

from left	$f(x)$	from right	$f(x)$
1	3	3	5
1.5	3.5	2.5	4.5
1.9	3.9	2.1	4.1
1.99	3.99	2.01	4.01
1.999	3.999	2.001	4.001
1.9999	3.9999	2.0001	4.0001

Table 2.2.1: outputs from both sides.

From both sides the values of

f(x)

close in on 4 — even though

f(2)=\tfrac{0}{0}

has no value. The graph heads toward 4 without ever reaching it.

The Limit of a Function · bookSHelf Calculus§2.2

Group handout — write down what you did

Name the method — handout

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. Write down exactly what your group did, step by step — then name and define your method.

Write the steps. Think about:

Which $x$ -values did you choose?
What did you compute at each?
How did you decide on the final value?
How did you make sure both sides agreed?

The Limit of a Function · bookSHelf Calculus§2.2

The method, named and defined

Name the method

You found where $f$ was heading as $x \to 2$ — without ever using $x = 2$ . Written out, the recipe is:

Choose $x$ -values closing in on the point from the left.
And matching values from the right.
Compute $f(x)$ at each one.
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 question driving this section

You pinned down where one function was headed near one point. Here is the question that drives the rest of the section:

How can we describe where a function is headed near a point — even when we can't, or shouldn't, plug the point in?

The hole at $x = 2$ is only the first way a point can't be trusted. By the end you'll have a name for every way a function can behave as $x$ approaches — and every way that behavior can fail.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.1 — the central idea of calculus

What a limit is

Definition 2.2.1 — Intuitive definition of a limit

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.

An open interval containing a, with a shaded neighborhood band

Figure 2.2.2: the limit reports what the neighborhood (the shaded band around $a$ ) 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 $\displaystyle\lim_{x \to 2} f(x) = 4$ .

The Limit of a Function · bookSHelf Calculus§2.2

Why the hole — and the big idea

Heading toward, not arriving at

Factor and cancel:

\frac{x^2-4}{x-2}=\frac{(x-2)(x+2)}{x-2}=x+2 \quad\text{for } x \ne 2.

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

Like reading which way a friend is heading through a crowd, even when you cannot see exactly where they stand — a limit reads where a function is headed, not where it is.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.2 — estimating a limit from a table

Example 2.2.1 — a limit from a table

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

Table 2.2.2: both sides march toward 1.

Figure 2.2.3: $y=\dfrac{\sin x}{x}$ near $x=0$ .

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

The Limit of a Function · bookSHelf Calculus§2.2

Try it now 2.2.1 — your turn

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

Simplify first: $\dfrac{\frac{1}{x}-1}{x-1}=\dfrac{-(x-1)}{x(x-1)}=-\dfrac{1}{x}$ for $x \ne 1$ .

$x$	$f(x)$	$x$	$f(x)$
0.9	−1.1111	1.1	−0.9091
0.99	−1.0101	1.01	−0.9901
0.999	−1.0010	1.001	−0.9990

Table 2.2.3: both columns march toward $-1$ .

Answer

$\displaystyle\lim_{x \to 1}\frac{\frac{1}{x}-1}{x-1}=-1.$

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.3 — estimating a limit from a graph

Example 2.2.3 — a limit from a graph

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:

\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.

Graph of g(x) with the curve approaching 3 but a marked point at 4

Figure 2.2.4: curve heads to 3; the marked point sits at 4.

The Limit of a Function · bookSHelf Calculus§2.2

From estimates to building blocks

Guesswork, then proof

Tables and graphs are quick, but they rest on guesswork — you are reading a trend, not proving a value. The next section builds algebraic limit laws that compute limits exactly.

Before we leave the informal stage, two limits are so basic that every one of those laws will lean on them.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.3 — the atomic building blocks

Theorem 2.2.1 — two important limits

Theorem 2.2.1 — Two basic limits

For any real number $a$ and constant $c$ :

\lim_{x \to a} x = a \qquad\text{and}\qquad \lim_{x \to a} c = c.

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

The Limit of a Function · bookSHelf Calculus§2.2

Two ways a limit can fail

When there is no single number

For a two-sided limit to exist, $f(x)$ must approach a single real number from both sides. It can miss in two very different ways:

Oscillation

$f$ bounces around forever without settling.

Blow-up

$f$ shoots off toward $\pm\infty$ , a vertical wall.

Both get labeled DNE for now — but the rest of the section gives each its own precise vocabulary.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.4 — a limit that fails by oscillation

Example 2.2.4 — $\displaystyle\lim_{x \to 0}\sin\!\left(\tfrac{1}{x}\right)$

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

\lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)\ \text{DNE.}

Graph of sin(1/x) oscillating ever faster near x=0

Figure 2.2.5: $y=\sin(1/x)$ oscillates without bound near 0.

The Limit of a Function · bookSHelf Calculus§2.2

When "DNE" hides the real story

Not every failure is a wander

Saying "the limit does not exist" sometimes throws away useful information.

$\sin(1/x)$ genuinely wandered — nothing more to say. But recall $g(x)=\dfrac{|x-2|}{x-2}$ : it settles cleanly on $-1$ from the left and $+1$ from the right.

The two-sided limit fails, yet each side is perfectly well-behaved. That deserves its own notation.

Figure 2.2.6: $g(x)=\dfrac{|x-2|}{x-2}$ — clean on each side, jump at $x=2$ .

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.5 — a name for each side

Definition 2.2.2 — one-sided limits

Definition 2.2.2 — One-sided limits

From the left. If $f(x)$ approaches $L$ as $x\ (x \lt a)$ approaches $a$ :

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

From the right. If $f(x)$ approaches $L$ as $x\ (x \gt a)$ approaches $a$ :

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

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 $\displaystyle\lim_{x \to 2^+} g(x) = 1$ .

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.5 — one-sided limits of a piecewise function

Example 2.2.5 — evaluating one-sided limits

f(x)=\begin{cases} x+1 & \text{if } x \lt 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.

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

Figure 2.2.7: the branches meet at different heights — a jump.

The Limit of a Function · bookSHelf Calculus§2.2

Try it now 2.2.4 — your turn

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

Factor: $|x^2-4| = |x-2|\,|x+2|$ .

For $x \gt 2$ : $\quad \dfrac{|x^2-4|}{x-2} \to |x+2| = 4$ .
For $x \lt 2$ : $\quad \dfrac{|x^2-4|}{x-2} \to -|x+2| = -4$ .

\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.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.5 — the handshake condition

Theorem 2.2.2 — one-sided and two-sided

Theorem 2.2.2 — Relating one-sided and two-sided limits

\lim_{x \to a} f(x) = L \;\iff\; \lim_{x \to a^-} f(x) = L \ \text{ and } \ \lim_{x \to a^+} f(x) = L.

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.

The Limit of a Function · bookSHelf Calculus§2.2

Back to the blow-up

The failure that never settles

Every function so far stayed bounded near its trouble point. But the third opening function never settles:

h(x)=\frac{1}{(x-2)^2}\ \longrightarrow\ +\infty \quad\text{as } x \to 2.

This is the blow-up failure we flagged earlier. Rather than write "DNE", we record how it fails — with the symbol $\infty$ .

Figure 2.2.8: $h(x)=\dfrac{1}{(x-2)^2}$ blows up at $x=2$ .

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.6 — naming the blow-up

Infinite limits & vertical asymptotes

Definition 2.2.3 — Infinite limits

If $f(x)$ increases without bound as $x \to a$ , we write $\displaystyle\lim_{x \to a} f(x) = +\infty$ ; if it decreases without bound, $\displaystyle\lim_{x \to a} f(x) = -\infty$ . The same applies to the one-sided versions $x \to a^-$ and $x \to a^+$ . Writing $\pm\infty$ records how the limit fails — infinity describes behavior, not a real-number value.

Definition 2.2.4 — Vertical asymptote

The line $x = a$ is a vertical asymptote of $f(x)$ if any one of $\displaystyle\lim_{x \to a^-} f(x) = \pm\infty$ , $\displaystyle\lim_{x \to a^+} f(x) = \pm\infty$ , or $\displaystyle\lim_{x \to a} f(x) = \pm\infty$ holds. Example: $h(x)=\dfrac{1}{(x-2)^2}$ has $\displaystyle\lim_{x \to 2} h(x) = +\infty$ , so $x = 2$ is a vertical asymptote.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.6 — recognizing an infinite limit

Example 2.2.6 — the 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 sides run off in opposite directions.

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

Graph of 1/x with a vertical asymptote at x=0

Figure 2.2.9: $y=\dfrac{1}{x}$ near $x=0$ .

The sides blow up in opposite directions, so $\displaystyle\lim_{x \to 0}\frac{1}{x}$ DNE.

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.6 — the even/odd rule

Theorem 2.2.3 — infinite limits from $\dfrac{1}{(x-a)^n}$

Theorem 2.2.3 — Infinite limits for positive integers $n$

If $n$ is even:

\lim_{x \to a}\frac{1}{(x-a)^n}=+\infty.

If $n$ is odd:

\lim_{x \to a^+}\frac{1}{(x-a)^n}=+\infty,\quad \lim_{x \to a^-}\frac{1}{(x-a)^n}=-\infty.

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

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.6 — finding a vertical asymptote

Example 2.2.7 — $f(x)=\dfrac{1}{(x+3)^4}$ at $x = -3$

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

Even $n$ means both sides shoot to $+\infty$ :

\lim_{x \to -3^-} f(x) = \lim_{x \to -3^+} f(x) = \lim_{x \to -3} f(x) = +\infty.

Vertical asymptote

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

The Limit of a Function · bookSHelf Calculus§2.2

Try it now 2.2.6 — your turn

$f(x)=\dfrac{1}{(x-2)^3}$ at $x = 2$

$n = 3$ is odd, so the side limits go to opposite infinities:

\lim_{x \to 2^-}\frac{1}{(x-2)^3}=-\infty,\qquad \lim_{x \to 2^+}\frac{1}{(x-2)^3}=+\infty.

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 $x = 2$ .

The Limit of a Function · bookSHelf Calculus§2.2

§2.2.6 — four behaviors on one graph

Example 2.2.8 — read each behavior off the 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.

Graph of f with a smooth point, a hole, a jump, and a vertical asymptote

Figure 2.2.10: one graph, four kinds of behavior.

2.2

The Limit of a Function · bookSHelf Calculus§2.2

§2.2 — conclusions

The question, answered

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

Table 2.2.5: the four verdicts, one per feature.

The same four-behavior graph, summarizing the section

Figure 2.2.10 (repeated): the four features together.

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. Next: §2.3 The Limit Laws. Back to start.

The Limit of a Function

Objectives

Closing in on x=2x = 2x=2

Name the method — handout

Name the method

What a limit is

Heading toward, not arriving at

Example 2.2.1 — a limit from a table

Estimate lim⁡x→11x−1x−1\displaystyle\lim_{x \to 1}\frac{\frac{1}{x}-1}{x-1}x→1lim​x−1x1​−1​

Example 2.2.3 — a limit from a graph

Guesswork, then proof

Theorem 2.2.1 — two important limits

When there is no single number

Oscillation

Blow-up

Example 2.2.4 — lim⁡x→0sin⁡ ⁣(1x)\displaystyle\lim_{x \to 0}\sin\!\left(\tfrac{1}{x}\right)x→0lim​sin(x1​)

Not every failure is a wander

Definition 2.2.2 — one-sided limits

Example 2.2.5 — evaluating one-sided limits

One-sided limits of ∣x2−4∣x−2\dfrac{|x^2-4|}{x-2}x−2∣x2−4∣​ at x=2x = 2x=2

Theorem 2.2.2 — one-sided and two-sided

The failure that never settles

Infinite limits & vertical asymptotes

Example 2.2.6 — the limits of 1x\dfrac{1}{x}x1​ at x=0x = 0x=0

Theorem 2.2.3 — infinite limits from 1(x−a)n\dfrac{1}{(x-a)^n}(x−a)n1​

Example 2.2.7 — f(x)=1(x+3)4f(x)=\dfrac{1}{(x+3)^4}f(x)=(x+3)41​ at x=−3x = -3x=−3

f(x)=1(x−2)3f(x)=\dfrac{1}{(x-2)^3}f(x)=(x−2)31​ at x=2x = 2x=2

Example 2.2.8 — read each behavior off the graph

The question, answered

Closing in on $x = 2$

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

Example 2.2.4 — $\displaystyle\lim_{x \to 0}\sin\!\left(\tfrac{1}{x}\right)$

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

Example 2.2.6 — the limits of $\dfrac{1}{x}$ at $x = 0$

Theorem 2.2.3 — infinite limits from $\dfrac{1}{(x-a)^n}$

Example 2.2.7 — $f(x)=\dfrac{1}{(x+3)^4}$ at $x = -3$

$f(x)=\dfrac{1}{(x-2)^3}$ at $x = 2$