🎯⭐ INTERACTIVE LESSON

One-Sided Limits

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One-Sided Limits - Complete Interactive Lesson

Part 1: Notation and Idea

👈👉 One-Sided Limits — Notation and Idea

Part 1 of 4 — Introducing $x \to a^-$ and $x \to a^+$

Topics in This Part

Section
Why We Need One-Sided Limits
🔑 The $-$ and $+$ Superscript Notation
Reading the Symbols Aloud

🔑 Why this matters: Some functions behave very differently from the left vs. the right. One-sided limits let us describe each side separately.

💡 Why We Need One-Sided Limits

Look at $f(x) = \dfrac{|x|}{x}$ .

For $x < 0$ : , so .

🔑 The $-$ and $+$ Superscript Notation

Notation	Meaning
$\lim_{x \to a^-} f(x)$

🗣️ Reading the Symbols Aloud

Symbol	English
$\lim_{x \to 2^-} f(x) = 5$	"The limit of as approaches 2 from the left equals 5"

Concept Check 🎯

Translate the Notation 🧮

For $f(x) = \dfrac{|x|}{x}$ :

1) $\lim_{x \to 0^-} f(x) = ?$ (whole number with sign)

Part 2: Computing One-Sided Limits

🧮 Computing One-Sided Limits

Part 2 of 4 — Piecewise functions, $|x|$ , and the sign function

Topics in This Part

Section
Piecewise Functions
Absolute Value Functions
The Sign Function

🔑 Why this matters: The most common one-sided-limit problems involve splitting cases — you must know which formula to use on which side.

🧩 Piecewise Functions

Suppose $f (x) = {\begin{cases} x^{2} + 1 & if x < 2 \\ 3 x - 4 \end{cases}$

Part 3: The Existence Theorem

✅ The Two-Sided Existence Theorem

Part 3 of 4 — When does $\lim_{x \to a} f(x)$ exist?

Topics in This Part

Section
🔑 The Theorem
📝 Worked Examples
Why It Matters

🔑 Why this matters: This is the defining theorem connecting one-sided and two-sided limits. AP graders look for it explicitly in justifications.

🔑 The Existence Theorem

Part 4: When Only One Side Exists

⚠️ When Only ONE Side Exists

Part 4 of 4 — Domain restrictions and one-sided endpoints

Topics in This Part

Section
Why Only One Side Might Exist
📝 Square Roots and Endpoints
The Two-Sided Limit at a Domain Endpoint

🔑 Why this matters: AP loves to ask about $\sqrt{x}$ at and similar functions where one side is .

For

x > 0

|x| = x

, so

f(x) = +1

The two sides of $x = 0$ "want" different values. A two-sided limit can't capture this — we need symbols that describe each side separately.

⚠️ Trap: the superscript $-$ does NOT mean "negative." It means "from the left" (smaller values). $\lim_{x \to 5^-}$ uses $x = 4.9, 4.99, \ldots$ — all positive numbers.

2) $\lim_{x \to 0^+} f(x) = ?$ (whole number with sign)

f (x) = {x^{2} + 1 3 x - 4 if x < 2 if x \geq 2

To compute $\lim_{x \to 2^-} f(x)$ , use the formula for $x < 2$ : $\lim_{x \to 2^-} (x^2 + 1) = 2^2 + 1 = 5.$

To compute $\lim_{x \to 2^+} f(x)$ , use the formula for $x > 2$ : $\lim_{x \to 2^+} (3x - 4) = 3(2) - 4 = 2.$

The two sides disagree (5 vs. 2), so $\lim_{x \to 2} f(x)$ DNE — jump.

💡 Strategy: identify which "piece" applies on each side, then substitute.

➖➕ Absolute Value Functions

Rewrite $|x - a|$ as a piecewise function: $|x - a| = \begin{cases} -(x - a) & \text{if } x < a \\ (x - a) & \text{if } x \ge a \end{cases}$

Worked example: $\displaystyle \lim_{x \to 3} \frac{|x - 3|}{x - 3}$ .

Left ( $x \to 3^-$ ): $|x - 3| = -(x - 3)$ , so the ratio is .

🔑 Whenever you see $|x - a|$ near $x = a$ , split into cases. Almost always you get $\pm 1$ on the two sides.

🔺 The Sign Function

$\operatorname{sgn}(x) = \begin{cases} -1 & x < 0 \\ 0 & x = 0 \\ +1 & x > 0 \end{cases}$

$\lim_{x \to 0^-} \operatorname{sgn}(x) = -1$
$\lim_{x \to 0^{+}} sgn$

💡 The value $\operatorname{sgn}(0) = 0$ doesn't change the limit answers — limits ignore the value at the point.

Computation Check 🎯

Compute 🧮

For $g(x) = \dfrac{|x + 4|}{x + 4}$ :

1) $\lim_{x \to -4^-} g(x) = ?$

2) $\lim_{x \to -4^+} g(x) = ?$

3) Does $\lim_{x \to -4} g(x)$ exist? yes or no:

$\displaystyle \lim_{x \to a} f(x) = L$ if and only if $\displaystyle \lim_{x \to a^-} f(x) = L$ and $\displaystyle \lim_{x \to a^+} f(x) = L$ .

Three things must all hold:

The left-hand limit exists.
The right-hand limit exists.
They agree on a single value $L$ .

If any one fails, the two-sided limit DNE.

💡 Notice: the value $f(a)$ itself does not enter the theorem at all. The theorem is about the two arms of the function — not the dot.

📝 Worked Examples

Example 1. $f(x) = x^2$ . $\lim_{x \to 3^-} f = 9$ and $\lim_{x \to 3^+} f = 9$ . Both agree → $\lim_{x \to 3} f = 9$ . ✅

Example 2. Piecewise $f$ with left limit 4 and right limit 4 at $x = 1$ , but $f(1) = 100$ . Both one-sided limits agree on 4 → $\lim_{x \to 1} f = 4$ . ✅ (The value doesn't change this.)

Example 3. Piecewise $f$ with left limit 2 and right limit 6 at $x = 0$ . Disagreement → $\lim_{x \to 0} f$ DNE.

Example 4. $f(x) = 1/x$ at $x = 0$ : left limit $= -\infty$ , right limit $= +\infty$ . Neither one-sided limit exists as a finite value, so the two-sided limit DNE.

🎯 Why It Matters

The theorem is the official tool for two of the most common AP tasks:

Task	How the theorem helps
Show a limit exists	Verify both one-sided limits exist and equal each other
Show a limit DNE	Show the one-sided limits disagree, OR show one is $\pm\infty$
Continuity check	Continuity at $a$ requires (limit exists) AND (limit = $f(a)$ ) — first half is the existence theorem

🔑 On free-response: when justifying continuity or DNE, cite the existence theorem by name or by structure.

Theorem Check 🎯

Apply the Theorem 🔽

🤔 Why Only One Side Might Exist

If a function is only defined on one side of $x = a$ , you can only approach $a$ from that side. The other one-sided limit doesn't exist (because the function values don't exist there).

Function	Domain near 0	Limits at 0
$\sqrt{x}$	$x \ge 0$ only	$\lim_{x \to 0^+} \sqrt{x} = 0$ ; left limit doesn't make sense
$\ln x$	$x > 0$ only	$\lim_{x \to 0^+} \ln x = -\infty$ ; left limit doesn't make sense
$\sqrt{x - 5}$	$x \ge 5$	; left limit not defined

📝 Worked Example: $\sqrt{x - 3}$ at $x = 3$

The expression $\sqrt{x - 3}$ requires $x \ge 3$ .

$\lim_{x \to 3^+} \sqrt{x - 3} = \sqrt{0} = 0$ . ✅ Defined.

💡 In AP-style problems, the answer here is "the right-hand limit equals 0; the left-hand limit does not exist."

⚖️ The Two-Sided Limit at a Domain Endpoint

What is $\lim_{x \to 3} \sqrt{x - 3}$ ?

By the strict existence theorem, the two-sided limit DNE because the left-hand limit doesn't exist.

⚠️ AP convention exception: for the purposes of continuity from the right at an endpoint, AP textbooks often define $\lim_{x \to 3} \sqrt{x - 3} = 0$ by treating the right-hand limit as the relevant one. Read the problem context: most AP problems will ask explicitly for , sidestepping the technicality.

Endpoint Reasoning 🎯

Right (

x \to 3^+

|x - 3| = x - 3

, so the ratio is

+1

Two-sided limit DNE.

lim_{x \to 0^{+}} sgn (x) = + 1

Two-sided

\lim_{x \to 0} \operatorname{sgn}(x)

DNE.

lim_{x \to 5^{+}} x - 5 = 0

\lim_{x \to 3^-} \sqrt{x - 3}

: for

x < 3

, the radicand is negative, so

\sqrt{x - 3}

isn't a real number. Left-hand limit doesn't exist.

One-Sided Limits

One-Sided Limits - Complete Interactive Lesson

Part 1: Notation and Idea

👈👉 One-Sided Limits — Notation and Idea

Topics in This Part

💡 Why We Need One-Sided Limits

🔑 The −-− and +++ Superscript Notation

🗣️ Reading the Symbols Aloud

Part 2: Computing One-Sided Limits

🧮 Computing One-Sided Limits

Topics in This Part

🧩 Piecewise Functions

Part 3: The Existence Theorem

✅ The Two-Sided Existence Theorem

Topics in This Part

🔑 The Existence Theorem

Part 4: When Only One Side Exists

⚠️ When Only ONE Side Exists

Topics in This Part

➖➕ Absolute Value Functions

🔺 The Sign Function

📝 Worked Examples

🎯 Why It Matters

🤔 Why Only One Side Might Exist

📝 Worked Example: x−3\sqrt{x - 3}x−3​ at x=3x = 3x=3

⚖️ The Two-Sided Limit at a Domain Endpoint

🔑 The $-$ and $+$ Superscript Notation

📝 Worked Example: $\sqrt{x - 3}$ at $x = 3$