🎯⭐ INTERACTIVE LESSON

Limits at Infinity

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Limits at Infinity - Complete Interactive Lesson

Part 1: Limits at Infinity & Horizontal Asymptotes

♾️ Limits at Infinity — End Behavior

Part 1 of 4 — What $\lim_{x \to \pm\infty}$ means

Topics in This Part

Section
The Idea of $x \to \pm\infty$
🔑 Horizontal Asymptotes
Quick Reference Table

🔑 Why this matters: End behavior governs horizontal asymptotes — a foundational topic for both limits and integration later.

💡 What Does $\lim_{x \to \infty} f(x) = L$ Mean?

As $x$ gets arbitrarily large, $f(x)$ gets arbitrarily close to .

🔑 Horizontal Asymptotes

The line $y = L$ is a horizontal asymptote of $f$ if $\lim_{x \to \infty} f(x) = L$ .

📋 Quick Reference

Function	$x \to +\infty$	$x \to -\infty$
$1/x$	$0$

End-Behavior Vocab 🎯

Compute Limits at Infinity 🧮

1) $\lim_{x \to \infty} \dfrac{5}{x} = ?$

Part 2: Rational Functions Compare Degrees

📊 Rational Functions at Infinity

Part 2 of 4 — Compare degrees

Topics in This Part

Section
🔑 The Three-Case Rule
Why It Works (Divide-By-Highest-Power)
Examples

🔑 Why this matters: Almost every AP "limit at infinity" problem on a rational function is solved by the degree-comparison rule.

🔑 The Three-Case Rule

For $\lim_{x \to \pm\infty} \dfrac{p(x)}{q(x)}$ with degrees , :

Part 3: Radicals at Infinity

√ Radicals at Infinity

Part 3 of 4 — Be careful with $|x|$ when $x \to -\infty$

Topics in This Part

Section
The "Effective Degree" of a Radical
🔑 The $\sqrt{x^2} = \lvert x \rvert$ Trap

Part 4: Exponentials, Logs, and DNE Cases

🌐 Other Cases — Exponentials, Logs, and Oscillation

Part 4 of 4 — Beyond rationals

Topics in This Part

Section
Exponentials Win the Growth Race
Logs Lose the Growth Race
When Limits at Infinity DNE

🔑 Why this matters: Many limit-at-infinity problems involve $e^x$ , $\ln x$ , or oscillating functions. Each has its own rules.

🚀 Exponentials Beat Polynomials

For any positive : grows faster than . So

We are not plugging $\infty$ into $f$ — $\infty$ isn't a number. We're asking: as $x$ marches off to the right, where is $f$ headed?

Same idea for $\lim_{x \to -\infty} f(x) = L$ — march to the left.

💡 The two ends are independent: $f$ might approach different limits as $x \to \infty$ and as $x \to -\infty$ .

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

A function can have:

0 horizontal asymptotes (e.g., $f(x) = x^2$ — both ends $\to +\infty$ )
1 horizontal asymptote (e.g., $f(x) = e^x$ — left end $\to 0$ , right end $\to \infty$ , so $y = 0$ on the left only)
2 horizontal asymptotes (e.g., $f(x) = \arctan x$ — $-\pi/2$ on the left, $\pi/2$ on the right)

⚠️ A function can cross a horizontal asymptote — it just can't stay away from it forever.

lim_{x \to - \infty} e^{x} = ?

3) $\lim_{x \to \infty} \arctan x = ?$ (use pi/2)

Case	Result
$m < n$ (bottom heavier)	Limit $= 0$
$m = n$ (equal degrees)	Limit $= \dfrac{\text{leading coef of } p}{\text{leading coef of } q}$
$m > n$ (top heavier)	Limit $= \pm\infty$ (sign depends on leading coefficients and direction) — DNE as a finite limit

💡 Memorize this. It's the single most useful rule in this section.

🛠️ Why It Works — Divide By Highest Power

Take $\lim_{x \to \infty} \dfrac{3x^2 + 5x}{2x^2 - 7}$ . Divide top and bottom by $x^2$ :

$\dfrac{3 + 5/x}{2 - 7/x^2}$

As $x \to \infty$ , $5/x \to 0$ and $7/x^2 \to 0$ :

$\dfrac{3 + 0}{2 - 0} = \dfrac{3}{2}$

The non-leading terms become negligible, leaving just the ratio of leading coefficients.

📝 Examples

Limit	Case	Answer
$\lim_{x \to \infty} \dfrac{4x + 1}{x^2 + 3}$	$m < n$	$0$
$\lim_{x \to \infty} \dfrac{6x^3 - 2x}{3x^3 + 5}$
$\lim_{x \to \infty} \dfrac{x^2}{x + 1}$
$\lim_{x \to -\infty} \dfrac{x^3 + 1}{x^2}$

Apply the Three-Case Rule 🎯

Compute 🧮

1) $\lim_{x \to \infty} \dfrac{8x^3 - 5}{4x^3 + x} = ?$

2) $\lim_{x \to \infty} \dfrac{3x + 1}{x^2 - 7} = ?$

3) $\lim_{x \to -\infty} \dfrac{6x^2 - 11}{2x^2 + 5} = ?$

🔑 Why this matters: A square root contributes "half a degree." And the sign on $x$ matters when $x$ is negative.

📐 Effective Degree

For end behavior, $\sqrt{x^2}$ behaves like $x$ in size — it has "effective degree 1." More generally, $\sqrt{x^{2k}}$ has effective degree $k$ .

Example. $\lim_{x \to \infty} \dfrac{\sqrt{x^2 + 1}}{x + 1}$ .

Numerator effective degree 1; denominator degree 1 → equal.
Leading behavior: numerator $\sim x$ , denominator $\sim x$ . Ratio $\to 1$ .

So this limit is $1$ as $x \to \infty$ .

⚠️ The $\sqrt{x^2} = |x|$ Trap

The identity is: $\sqrt{x^2} = |x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$

So when $x \to -\infty$ , $\sqrt{x^2} = -x$ (positive), . This sign change matters.

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

Divide top and bottom by $|x| = -x$ (since $x \to -\infty$ ):

$\dfrac{\sqrt{1 + 1/x^2}}{(x + 1)/(-x)} = \dfrac{\sqrt{1 + 1/x^2}}{-1 - 1/x}$

As $x \to -\infty$ : numerator $\to \sqrt{1} = 1$ , denominator .

Answer: $\dfrac{1}{-1} = -1$ .

💡 Compare: as $x \to +\infty$ the same expression gives $+1$ . The two ends differ in sign.

🚀 Shortcut: Look at Leading Behavior

For $\sqrt{ax^2 + \ldots}$ as $x \to \pm\infty$ :

$x \to +\infty$ : $\sqrt{ax^2 + \ldots} \sim \sqrt{a}\,x$

This shortcut handles most AP problems quickly.

Sign-Aware Limits 🎯

Practice 🧮

1) $\lim_{x \to \infty} \dfrac{\sqrt{9x^2 + x}}{x + 5} = ?$

2) $\lim_{x \to -\infty} \dfrac{\sqrt{9x^2 + x}}{x + 5} = ?$

\lim_{x \to \infty} \dfrac{x^n}{e^x} = 0, \qquad \lim_{x \to \infty} \dfrac{e^x}{x^n} = +\infty.

Example. $\lim_{x \to \infty} \dfrac{x^{100}}{e^x} = 0$ . (Even with a huge polynomial, the exponential wins.)

For $x \to -\infty$ : $e^x \to 0$ , so things like $x^2 e^x \to 0$ (still exponential dominates the polynomial).

🐢 Logs Lose to Polynomials

For any positive $n$ : $\ln x$ grows slower than $x^n$ .

$\lim_{x \to \infty} \dfrac{\ln x}{x^n} = 0, \qquad \lim_{x \to \infty} \dfrac{x^n}{\ln x} = +\infty.$

Growth hierarchy (slowest to fastest, going to $+\infty$ ): $\ln x \;\ll\; x^{1/2} \;\ll\; x \;\ll\; x^2 \;\ll\; e^x$

🚫 When the Limit at Infinity DNE

Some functions don't settle to a value or to $\pm\infty$ at infinity:

$\sin x$ , $\cos x$ as $x \to \infty$ — oscillate forever between $-1$ and $1$ . DNE.
$x \sin x$ — amplitude grows, oscillates between $\pm\infty$ . DNE.

But:

$\dfrac{\sin x}{x} \to 0$ as $x \to \infty$ (Squeeze Theorem: bounded numerator, growing denominator).

💡 Bounded over growing gives 0. Growing × bounded oscillation fails to converge.

Beyond Rationals 🎯

x \to -\infty

\sqrt{ax^2 + \ldots} \sim -\sqrt{a}\,x

(because we need a positive output and

x

is negative)

Limits at Infinity

Limits at Infinity - Complete Interactive Lesson

Part 1: Limits at Infinity & Horizontal Asymptotes

♾️ Limits at Infinity — End Behavior

Topics in This Part

💡 What Does lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = Llimx→∞​f(x)=L Mean?

🔑 Horizontal Asymptotes

📋 Quick Reference

Part 2: Rational Functions Compare Degrees

📊 Rational Functions at Infinity

Topics in This Part

🔑 The Three-Case Rule

Part 3: Radicals at Infinity

√ Radicals at Infinity

Topics in This Part

Part 4: Exponentials, Logs, and DNE Cases

🌐 Other Cases — Exponentials, Logs, and Oscillation

Topics in This Part

🚀 Exponentials Beat Polynomials

🛠️ Why It Works — Divide By Highest Power

📝 Examples

📐 Effective Degree

⚠️ The x2=∣x∣\sqrt{x^2} = |x|x2​=∣x∣ Trap

🚀 Shortcut: Look at Leading Behavior

🐢 Logs Lose to Polynomials

🚫 When the Limit at Infinity DNE

💡 What Does $\lim_{x \to \infty} f(x) = L$ Mean?

⚠️ The $\sqrt{x^2} = |x|$ Trap