Calculus I · Limits and Continuity · lesson

What a Limit Means

Visual study stop

Read the picture before the symbols

Pause at each graph long enough to say what the input is doing, what the output is doing, and which feature supports the next mathematical decision.

Graph of y equals x plus 2 with a removable hole at (2, 4). — **Separate the target point from nearby behavior**
Trace the curve toward the open circle from both directions before looking at whether the function is defined at the target. The gathering height determines the limit; the target point does not.

Concept

Learning objectives

Interpret $\lim_{x\to a}f(x)=L$ in words; distinguish the input being approached from the output being approached; estimate a limit numerically and graphically.

The Basic Language of Limits

Concept

When we say "the limit is 7," we are not saying the input becomes 7. We are saying the output approaches 7 while the input approaches some specified number.

Definition

We write

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

when the values of $f(x)$ can be made as close as desired to $L$ by taking $x$ sufficiently close to $a$ , with $x\ne a$ .

Read

\lim_{x\to3}f(x)=7

"The limit of $f(x)$ as $x$ approaches $3$ is $7$ ."

The roles are:

\underbrace{x}_{\text{changing input}} \to \underbrace{3}_{\text{input target}}, \qquad \underbrace{f(x)}_{\text{changing output}} \to \underbrace{7}_{\text{output target}}.

Guided walkthrough

A limit with no hole

Let $f(x)=x+2$ . What happens to $f(x)$ as $x\to3$ ?

Show worked solution

If $x$ is close to $3$ , then $x+2$ is close to $5$ :

Reference table
$x$	$x+2$
$2.9$	$4.9$
$2.99$	$4.99$
$3.01$	$5.01$
$3.1$	$5.1$

Therefore,

\boxed{\lim_{x\to3}(x+2)=5}.

This limit is direct because the function has no break at $x=3$ . Later we will call this continuity.

Interactive checklimit-continuous-01

Evaluate $\lim_{x\to3}(x^2+1)$ .

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Show hint

This polynomial has no break at $x=3$ .

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After the explanation

Use the section idea

Reading lens

What are nearby outputs doing as the input approaches the target from both sides?

Mental model

Imagine tightening a window around the target input and watching where all nearby outputs are forced to gather.

Decision

Read the left-hand and right-hand behavior separately first; combine them only after both sides approach the same output.

Common trap

The function value at the target can be missing or deliberately moved, so never substitute a plotted dot for evidence from both sides.

Check yourself

You understand the section when you can explain a limit from a graph, table, and sentence without confusing it with the function value.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

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