🎯⭐ INTERACTIVE LESSON

Infinite Limits and Vertical Asymptotes

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Infinite Limits and Vertical Asymptotes - Complete Interactive Lesson

Part 1: Notation and Meaning

📈 Infinite Limits — When Functions Blow Up

Part 1 of 4 — Notation and meaning

Topics in This Part

Section
What " $\lim = \infty$ " Really Means
🔑 Infinite Limits vs. Limits at Infinity
Notation and Conventions

🔑 Why this matters: Infinite limits describe vertical asymptotes — a key feature of any rational function.

💡 What " $\lim_{x \to a} f(x) = +\infty$ " Means

As $x$ approaches $a$ , the values of grow without bound (positive direction).

⚖️ Infinite Limits vs. Limits at Infinity

Two different concepts, similar names — don't confuse them.

Type	Notation	Means
Limit at infinity	$\lim_{x \to \infty} f(x)$	Behavior as $x$ goes to — describes asymptotes

📝 Notation and Conventions

Notation	Meaning
$\lim_{x \to a^+} f(x) = +\infty$	Approaching from the , shoots up

Notation Check 🎯

Identify 🧮

For $f(x) = \dfrac{1}{(x - 5)^2}$ :

1) (use , , , or a number)

Part 2: Diagnosing Infinite Limits

🔍 Diagnosing Infinite Limits

Part 2 of 4 — The "nonzero / zero" pattern

Topics in This Part

Section
🔑 The Nonzero-Over-Zero Test
Sign Analysis on Each Side
Worked Examples

🔑 Why this matters: Recognizing the $c/0$ pattern (nonzero $c$ ) tells you immediately you have a vertical asymptote — and a sign analysis tells you which infinity.

🔑 The Nonzero-Over-Zero Test

For $\lim_{x \to a} \frac{p (x)}{q (}$ :

Part 3: Vertical Asymptotes

📍 Vertical Asymptotes

Part 3 of 4 — Where do they live?

Topics in This Part

Section
Definition
🔑 Where to Find Them
Holes vs. Asymptotes (the $0/0$ vs. nonzero/0 distinction)

🔑 Why this matters: Vertical asymptotes are a graph feature; understanding them connects limits to function behavior.

📐 Definition

The vertical line $x = a$ is a vertical asymptote of $f$ if at least one of the following holds:

Part 4: Asymptotes Beyond Rationals

🌐 Asymptotes Beyond Rationals

Part 4 of 4 — Logs, tangent, and reading from a graph

Topics in This Part

Section
$\ln x$ at $x = 0$
$\tan x$ at $x = π / 2 + k$

This is not saying the limit exists in the usual sense — $\infty$ isn't a real number. It's a description of how the limit fails to be a finite number.

Same for $-\infty$ : $f(x)$ decreases without bound.

💡 In AP language, we sometimes say "the limit equals $+\infty$ " and sometimes say "the limit DNE (and the function $\to +\infty$ )". Both are accepted on the exam.

Quick test: is $\infty$ in the target (under the lim) or in the value (after the equal sign)?

Under the lim → limit at infinity (horizontal asymptote idea).
After the equal sign → infinite limit (vertical asymptote idea).

⚠️ If the two sides go to opposite infinities (e.g., left $\to -\infty$ , right $\to +\infty$ ), the two-sided limit DNE. We do not say it equals $\infty$ in that case.

\lim_{x \to 5^+} f(x) = ?

lim_{x \to 5^{+}} f (x) = ?

2) $\lim_{x \to 5^-} f(x) = ?$

3) $\lim_{x \to 5} f(x) = ?$

lim_{x \to a} \frac{p ( x )}{q ( x )}

Direct substitution gives	Conclusion
Number $L$	Limit is $L$ ✅
$0/0$	Indeterminate — factor/rationalize
Nonzero / 0	Vertical asymptote at $x = a$ — limit is $\pm\infty$ or DNE

The key is: if the denominator is approaching 0 but the numerator is approaching a nonzero number, the magnitude blows up.

💡 The next question is just which sign of infinity each side gives.

➕➖ Sign Analysis

Once you know "nonzero / zero," determine the sign of the result on each side of $a$ :

Numerator sign: plug $a$ into $p(x)$ — get the sign (positive or negative).
Denominator sign as $x \to a^+$ : is $q(x)$ slightly positive or slightly negative?
Denominator sign as $x \to a^-$ : same question.
Combine: like signs → $+\infty$ , opposite signs → $-\infty$ .

Worked example. $\lim_{x \to 2^+} \dfrac{1}{x - 2}$ .

Numerator: $1 > 0$ .
For $x$ slightly bigger than 2: $x - 2$ is small positive.
$\dfrac{+}{+} = +$ . So .

For $\lim_{x \to 2^-} \dfrac{1}{x - 2}$ : slightly , so , .

Sides disagree → two-sided limit DNE.

📝 Worked Examples

Limit	Numerator at $a$	Denominator on each side	Result
$\lim_{x \to 3^+} \dfrac{x}{x - 3}$	$+3$	small $+$	$+\infty$
$\lim_{x \to 3^-} \dfrac{x}{x - 3}$
$\lim_{x \to 0^+} \dfrac{-1}{x}$
$\lim_{x \to 0^-} \dfrac{-1}{x}$
$\lim_{x \to 1} \dfrac{1}{(x-1)^2}$

Sign Analysis 🔽

$\lim_{x \to a^+} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to a^-} f(x) = \pm\infty$ .

In other words: the function shoots off to $\pm\infty$ as $x$ approaches $a$ from at least one side.

💡 You don't need both sides to agree. A one-sided infinite limit is enough.

🔍 Where to Find Vertical Asymptotes

For a rational function $\dfrac{p(x)}{q(x)}$ :

Find zeros of $q(x)$ — these are the candidate asymptotes.
For each zero $a$ , check the numerator $p(a)$ :
- $p(a) \ne 0$ → vertical asymptote at $x = a$ .
- $p(a) = 0$ → it's a situation. Could be a (removable) or a vertical asymptote, depending on multiplicities.

Worked example. $f(x) = \dfrac{x + 1}{x^2 - 4} = \dfrac{x + 1}{(x - 2)(x + 2)}$ .

Denominator zeros: $x = 2$ and $x = -2$ .
Numerator at $x = 2$ : $3 \ne 0$ → asymptote at .

🕳️ Holes vs. Asymptotes

If both $p(a) = 0$ and $q(a) = 0$ , factor to figure out which it is.

Example 1 — hole. $f(x) = \dfrac{x^2 - 1}{x - 1} = \dfrac{(x-1)(x+1)}{x - 1} = x + 1$ for $x \ne 1$ . The factor cancels exactly. There's a hole at $x = 1$ , no asymptote. The limit there equals $2$ .

Example 2 — asymptote. $f(x) = \dfrac{x - 1}{(x - 1)^2} = \dfrac{1}{x - 1}$ . After cancelling, the denominator still has a remaining , so there's an asymptote.

💡 Rule: count multiplicities. If the denominator's power of $(x - a)$ is greater than the numerator's, you have an asymptote. Otherwise, a hole (or no issue).

Identify Asymptotes 🎯

Count Asymptotes 🧮

How many vertical asymptotes does each function have?

1) $f(x) = \dfrac{1}{x^2 - 9}$ → ?

2) $g(x) = \dfrac{x - 1}{x^2 - 1}$ → ? (think about cancellation)

3) $h(x) = \dfrac{x + 2}{(x - 1)(x - 5)}$ → ?

🔑 Why this matters: Vertical asymptotes appear in many functions, not just rationals.

📉 Natural Log at $x = 0$

The function $\ln x$ is defined for $x > 0$ and:

$\lim_{x \to 0^+} \ln x = -\infty$

So $x = 0$ is a vertical asymptote of $\ln x$ — but only on the right side (the function isn't defined for $x \le 0$ ).

Same for $\log_b x$ with any base $b > 1$ .

📐 Tangent's Asymptotes

$\tan x = \dfrac{\sin x}{\cos x}$ has vertical asymptotes wherever $\cos x = 0$ , i.e., at:

$x = \frac{\pi}{2} + k\pi \quad (k \in \mathbb{Z})$

At each one, the left and right limits go to opposite infinities.

Value	Left limit	Right limit
$x = \pi/2$	$+\infty$	$-\infty$
$x = 3\pi/2$

💡 The pattern repeats with period $\pi$ .

👁️ Reading Asymptotes from a Graph

A vertical asymptote appears as a vertical "cliff" on the graph: the curve rockets up or down as it approaches a vertical line $x = a$ .

Visual signs:

The curve rises (or falls) without bound near $x = a$ .
A vertical dashed line is often drawn at $x = a$ in textbook plots.
The function is undefined at $x = a$ .

⚠️ Don't confuse a hole (single missing point) with an asymptote (unbounded behavior). Holes look normal locally, just with one point missing; asymptotes show the function blowing up.

Apply the Ideas 🎯

Numerator at

x = -2

-1 \ne 0

→ asymptote at

x = -2

Two vertical asymptotes.

Infinite Limits and Vertical Asymptotes

Infinite Limits and Vertical Asymptotes - Complete Interactive Lesson

Part 1: Notation and Meaning

📈 Infinite Limits — When Functions Blow Up

Topics in This Part

💡 What "lim⁡x→af(x)=+∞\lim_{x \to a} f(x) = +\inftylimx→a​f(x)=+∞" Means

⚖️ Infinite Limits vs. Limits at Infinity

📝 Notation and Conventions

Part 2: Diagnosing Infinite Limits

🔍 Diagnosing Infinite Limits

Topics in This Part

🔑 The Nonzero-Over-Zero Test

Part 3: Vertical Asymptotes

📍 Vertical Asymptotes

Topics in This Part

📐 Definition

Part 4: Asymptotes Beyond Rationals

🌐 Asymptotes Beyond Rationals

Topics in This Part

➕➖ Sign Analysis

📝 Worked Examples

🔍 Where to Find Vertical Asymptotes

🕳️ Holes vs. Asymptotes

📉 Natural Log at x=0x = 0x=0

📐 Tangent's Asymptotes

👁️ Reading Asymptotes from a Graph

💡 What " $\lim_{x \to a} f(x) = +\infty$ " Means

📉 Natural Log at $x = 0$