Infinite Limits and Vertical Asymptotes

📈 Infinite Limits — When Functions Blow Up

Part 1 of 4 — Notation and meaning

Topics in This Part

🔑 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 $f(x)$ grow without bound (positive direction).

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.

⚖️ Infinite Limits vs. Limits at Infinity

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

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).

📝 Notation and Conventions

⚠️ 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.

Notation Check 🎯

Identify 🧮

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

1) $\lim_{x \to 5^+} f(x) = ?$ (use infinity, -infinity, DNE, or a number)

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

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

🔍 Diagnosing Infinite Limits

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

Topics in This Part

🔑 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 $\lim_{x \to a} \dfrac{p(x)}{q(x)}$:

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 $a$:

1. Numerator sign: plug $a$ into $p(x)$ — get the sign (positive or negative).
2. Denominator sign as $x \to a^+$: is $q(x)$ slightly positive or slightly negative?
3. Denominator sign as $x \to a^-$: same question.
4. 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 $\to +\infty$.

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

Sides disagree → two-sided limit DNE.

📝 Worked Examples

Diagnose 🎯

Sign Analysis 🔽

📍 Vertical Asymptotes

Part 3 of 4 — Where do they live?

Topics in This Part

🔑 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:

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

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 $\dfrac{p(x)}{q(x)}$:

1. Find zeros of $q(x)$ — these are the candidate asymptotes.
2. For each zero $a$, check the numerator $p(a)$:
   • $p(a) \ne 0$ → vertical asymptote at $x = a$.
   • $p(a) = 0$ → it's a $0/0$ situation. Could be a hole (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 $x = 2$.
• Numerator at $x = -2$: $-1 \ne 0$ → asymptote at $x = -2$.
• Two vertical asymptotes.

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

🌐 Asymptotes Beyond Rationals

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

Topics in This Part

🔑 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.

💡 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 🎯

Exit Check ✅