1.2 - Defining Limits and Using Limit Notation

1.2 Defining Limits and Using Limit Notation

Understanding Limits

A limit describes what value a function approaches as the input approaches a particular number. The key idea: we care about where the function is heading, not necessarily where it actually is.

Formal Notation

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

This reads: “the limit of

f(x)

x

approaches

a

equals

L

.”

What it means: as

x

values get closer and closer to

a

(from both the left and the right), the

y

values get closer and closer to

L

Two details make this precise:

$x$ gets close to $a$ but never equals $a$ . The limit only looks at inputs near $a$ , so the value $f(a)$ , or whether it even exists, has no effect on the limit. That’s why all three scenarios below have the same limit.

The limit has to be a single real number. $\lim_{x \to a} f(x) = L$ means we can make $f(x)$ as close to $L$ as we want just by taking $x$ close enough to $a$ . If no single real number works, the limit does not exist.

Not on the exam: the formal epsilon-delta definition of a limit (the exact, symbol-heavy version of “as close as we want”) isn’t tested on the AP Calculus AB or BC exam.

Three Key Scenarios

Scenario 1: limit equals the function value

1.2 - Defining Limits and Using Limit Notation

\lim_{x \to 3} f(x) = 2 \qquad f(3) = 2

x

approaches 3 from both directions,

f(x)

approaches 2. The function is also defined at

x = 3

, and its value there is 2. Limit and function value match.

Scenario 2: limit exists, function undefined

\lim_{x \to 3} f(x) = 2 \qquad f(3) = \text{undefined}

x

approaches 3 from both sides,

f(x)

still approaches 2. But at exactly

x = 3

there’s a hole in the graph (open circle), so the function has no value there. The limit exists even though the function value doesn’t.

Scenario 3: limit and function value differ

\lim_{x \to 3} f(x) = 2 \qquad f(3) = 5

x

approaches 3,

f(x)

approaches 2. But the actual value at

x = 3

is 5 (the filled dot above). The limit and function value are different, and that’s completely fine.

Practice Problems

$\lim_{x \to 2} f(x) = ?$

$f(-4) = ?$

$\lim_{x \to 4} f(x) = ?$

$f(4) = ?$

$f(2) = ?$

$\lim_{x \to -1} f(x) = ?$

$f(-1) = ?$

$\lim_{x \to -4} f(x) = ?$

Solutions

$\lim_{x \to 2} f(x) = -2$ . Approach $x = 2$ from both sides. From the left and right, the function heads toward $y = -2$ .

$f(-4) = 0$ . Look at where $x = -4$ . The filled dot is at $y = 0$ (ignore the open circle).

$\lim_{x \to 4} f(x) = 3$ . Both sides approach $y = 3$ .

$f(4) = 1$ . At exactly $x = 4$ , the filled dot is at $y = 1$ .

$f(2) = -2$ . At $x = 2$ , the filled dot shows $y = -2$ .

$\lim_{x \to -1} f(x) = -3$ . Approaching from both directions leads to $y = -3$ .

$f(-1) = \text{undefined}$ . There’s only an open circle at $x = -1$ , no filled dot, so the function doesn’t exist there.

$\lim_{x \to -4} f(x) = 1$ . From both sides, the function approaches $y = 1$ . Notice this is different from $f(-4) = 0$ .

Interpreting Limit Statements

If we have:

\lim_{x \to 5} g(x) = 12

We read it as: “as

x

approaches 5,

g(x)

approaches 12.”

Critical concept: a limit tells you what the function is approaching, not what the function equals at that point.

\lim_{x \to a} f(x) = L \quad \text{does NOT necessarily mean} \quad f(a) = L

One Limit, Three Representations

A limit can be shown graphically, numerically, or analytically. Here’s the same limit all three ways, using

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

near

x = 1

Graphically: the graph is the line $y = x + 1$ with a hole at $(1, 2)$ . Following the line toward $x = 1$ from either side leads to a height of 2.

Numerically: $f(0.9) = 1.9$ , $f(0.99) = 1.99$ , $f(1.01) = 2.01$ , and $f(1.1) = 2.1$ . The outputs close in on 2 from both sides.

Analytically: factor and cancel, $\frac{(x - 1)(x + 1)}{x - 1} = x + 1$ for $x \neq 1$ , so the limit is $1 + 1 = 2$ .

All three tell the same story:

\lim_{x \to 1} f(x) = 2

. Topics 1.3 to 1.8 go through each method in detail.

True or False

Question 1:

f(c) = \lim_{x \to c} f(x)

in all cases.

FALSE. Counterexample:

Here

\lim_{x \to c} f(x) = 3

but

f(c) = 7

Question 2:

f(c) \neq \lim_{x \to c} f(x)

in all cases.

FALSE. This is also false, because there are plenty of cases where they are equal. Counterexample:

Here

\lim_{x \to c} f(x) = 3

AND

f(c) = 3

Bottom line: the function value and the limit can be equal, but they don’t have to be. It depends on the function.

Key Takeaways

Limits describe approaching behavior, not necessarily actual values

We look at what happens as $x$ gets close to a value from both sides

Three possibilities:

Limit equals the function value (continuous)
Limit exists but the function is undefined (removable discontinuity)
Limit exists but differs from the function value (point discontinuity)

Open circles mean the function is undefined at that point

Filled dots show the actual function value at that point

The limit can exist even when the function value doesn’t

The same limit can be read from a graph, a table, or an equation

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