Calculus I · Limits and Continuity · lesson

Infinite Limits Explained

Concept

Learning objectives

Interpret infinite-limit notation; determine one-sided signs near a vertical asymptote; distinguish unbounded behavior from an ordinary finite limit.

Infinite Behavior and Asymptotes

Infinite Limits

A finite limit asks whether outputs approach one real number. An infinite limit describes outputs whose magnitude grows without bound.

Definition

Infinite-Limit Notation

The statement

\lim_{x\to a}f(x)=+\infty

means that $f(x)$ becomes arbitrarily large and positive as $x$ approaches $a$ .

The statement

\lim_{x\to a}f(x)=-\infty

means that $f(x)$ becomes arbitrarily negative as $x$ approaches $a$ .

Infinity is not a real number. The notation records a direction of unbounded growth.

Concept

Imagine a thermometer with no top. Saying the reading approaches $+\infty$ does not mean it arrives at a final number called infinity. It means that no matter how large a height you name, the reading eventually exceeds it near the target input.

Guided walkthrough

$1/x$ near zero

Consider $f(x)=1/x$ .

From the right of zero, use small positive numbers:

Reference table
$x$	$1/x$
$0.1$	$10$
$0.01$	$100$
$0.001$	$1000$

Thus,

\lim_{x\to0^+}\frac1x=+\infty.

From the left, use small negative numbers:

Reference table
$x$	$1/x$
$-0.1$	$-10$
$-0.01$	$-100$
$-0.001$	$-1000$

Thus,

\lim_{x\to0^-}\frac1x=-\infty.

The one-sided behaviors differ, so the ordinary two-sided limit does not exist.

Two panels comparing odd and even reciprocal powers near x = 1. — **Read the graph**
An odd power changes sign across the vertical asymptote; an even power remains positive on both sides.

After the explanation

Use the section idea

Reading lens

Is the function growing without bound near a finite input, or settling into end behavior as the input grows?

Mental model

Vertical asymptotes describe local blow-up near an excluded finite input; end-behavior asymptotes describe the long-run trend as inputs grow in magnitude.

Decision

Near a denominator zero, build a sign chart for each side; at infinity, compare dominant powers or divide to expose the lasting term.

Common trap

Do not merge positive and negative infinity, and remember that square roots produce absolute values when factoring a large squared input.

Check yourself

You understand the section when you can predict signs and asymptotes before doing detailed algebra, then verify them with the expression.

Source & rights

Original instruction with traceable references.

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