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What is a Limit?

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What is a Limit? - Complete Interactive Lesson

Part 1: The Intuitive Idea of a Limit

🎯 What Is a Limit?

Part 1 of 7 — The intuitive idea

Topics in This Part

Section
The Big Idea
📌 An Everyday Analogy
🔑 What a Limit Asks
Why $f(a)$ and $\lim_{x\to a} f(x)$ Are Different

🔑 Why this matters: Limits are the foundation of every other idea in calculus. The derivative, the definite integral, continuity, asymptotes — all are limits in disguise.

What You'll Master in Part 1

The intuitive ("approaching") meaning of a limit
The crucial distinction between what $f$ is doing near $a$ and what $f(a)$ equals
Why you cannot just "plug in" when the function misbehaves at the target point

💡 The Big Idea

A limit asks the question:

As $x$ gets arbitrarily close to a target value $a$ , what number is $f(x)$ getting arbitrarily close to?

We write this as

$\boxed{\lim_{x \to a} f(x) = L}$

🚗 An Everyday Analogy

Imagine you're driving toward a town named $a$ on the highway. As your odometer reading $x$ gets closer to $a$ , the elevation $f(x)$ of the highway approaches some specific value $L$ — the elevation of the town.

The limit is "what elevation will I reach as I get to town ?" — that's .

📌 $\lim_{x\to a} f(x)$ vs. $f(a)$

These are different questions.

Question	What it asks	Notation
What is the limit?

Concept Check 🎯

Numerical Sanity Check 🧮

The function $f(x) = \dfrac{x^2 - 9}{x - 3}$ is undefined at . Fill in the table to see what value the function approaches.

Match the Concept 🔽

Exit Check ✅

Part 2: Limit Notation

📝 Limit Notation

Part 2 of 7 — Reading and writing limits like a mathematician

Topics in This Part

Section
📌 The Standard Notation $\lim_{x \to a} f(x) = L$
🔑 One-Sided Notation: $x \to a^{-}$ vs

Part 3: When Does a Limit Exist?

✅ When Does a Limit Exist?

Part 3 of 7 — The "left = right" rule

Topics in This Part

Section
🔑 The Existence Theorem
Sided Limits Walkthrough
📌 Why "approach the same value" matters
Quick Diagnostic Procedure

🔑 Why this matters: Almost every "does this limit exist?" AP question reduces to checking whether the two one-sided limits agree.

🔑 The Existence Theorem

Theorem. $\lim_{x \to a} f(x) = L$ exists

Part 4: Limits That Do Not Exist

🚫 Limits That Do NOT Exist

Part 4 of 7 — The three classic failure modes

Topics in This Part

Section
📌 Failure Mode 1: Jump
📌 Failure Mode 2: Unbounded (blow-up)
📌 Failure Mode 3: Oscillation
Putting Them Side-by-Side

🔑 Why this matters: "Does the limit exist?" is one of the most common AP free-response prompts. You'll be expected to name the failure mode and explain why the limit doesn't exist.

1️⃣ Jump

The two one-sided limits exist but disagree.

Canonical example: $f(x) = \dfrac{|x|}{x}$ .

read aloud as "the limit of $f$ of $x$ as $x$ approaches $a$ equals $L$ ."

Word	What it means
Approaches	$x$ is getting closer and closer to $a$ — but $x \ne a$
Arbitrarily close	We can make the gap between $f(x)$ and $L$ as small as we like by choosing $x$ close enough to $a$
Equals $L$	The output is settling on a single value

💡 Limits are about a destination, not about the journey. The path $x$ takes to get to $a$ doesn't matter — only what $f(x)$ trends toward.

The value

f(a)

is "what is the elevation exactly at the town sign?" That might be missing (sign blown over), wrong (sign vandalized to say something silly), or different from

L

for any number of reasons.

The whole power of calculus comes from this separation: we can describe what's happening as you approach a point without needing the function to be defined or sensible at the point itself.

Three things that can happen

For the function $f(x) = \dfrac{x^2 - 4}{x - 2}$ :

Direct substitution gives $0/0$ , so $f(2)$ is undefined.
But near $x = 2$ , factor: $\dfrac{(x-2)(x+2)}{x-2} = x + 2$ for $x \neq 2$ .
So $\lim_{x \to 2} f(x) = 2 + 2 = 4$ , even though $f(2)$ doesn't exist.

💡 The limit can exist even when $f(a)$ does not. This is exactly why limits are the right tool for calculus.

1) $f(2.9) = ?$ (to 1 decimal place)

2) $f(2.99) = ?$ (to 2 decimal places)

3) $f(3.01) = ?$ (to 2 decimal places)

4) Based on your numerical evidence, $\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3} = ?$ (whole number)

🔑 Why this matters: Calculus is a language. Once you can fluently read $\lim_{x \to 3^+} f(x) = 5$ and translate it into "as $x$ approaches 3 from the right, $f(x)$ approaches 5," every theorem will be much easier to absorb.

📌 Standard Two-Sided Limit Notation

$\boxed{\lim_{x \to a} f(x) = L}$

Piece	Meaning
$\lim$	"the limit of"
$x \to a$	"as $x$ approaches $a$ "
$f(x)$

Read it aloud as: "the limit, as $x$ approaches $a$ , of $f(x)$ , equals $L$ ."

💡 The arrow $\to$ never means "equals." $x$ never actually reaches $a$ — it gets arbitrarily close.

🔑 One-Sided Limits

Sometimes the function behaves differently on each side of $a$ . We use a small $-$ or $+$ on the limit:

$\lim_{x \to a^-} f(x) \quad \text{(approach from the left)}$

$\lim_{x \to a^+} f(x) \quad \text{(approach from the right)}$

Symbol	Meaning	Mental picture
$x \to a^-$	$x$ slides up toward $a$ from values less than $a$

🔑 Key fact: The two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal: $\lim_{x \to a^{-}} f (x) = \lim_{x \to a^{+}} f (x) = L ⟺ \lim_{x \to}$

♾️ Limits Involving Infinity

Two completely different scenarios use the symbol $\infty$ :

1. Limits at infinity (end behavior of $f$ )

$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$

Asks: "what value is $f$ approaching as $x$ runs off to the right (or left) forever?"

2. Limits equal to infinity (vertical asymptotes)

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

Asks: "as $x$ approaches $a$ , does $f(x)$ grow without bound?" (Note: technically the limit does not exist in the usual sense — we use $\infty$ as a description of the unbounded behavior.)

Notation	Translation
$\lim_{x \to \infty} (1/x) = 0$	"as $x$ runs to the right forever, $1 /$ shrinks toward 0"

⚠️ Common Notation Pitfalls

❌ Wrong	✅ Right	Why
$\lim_{x = a} f(x)$	$\lim_{x \to a} f(x)$	$x$ approaches, never equals
$\lim f(x) = L$ (no $x \to$ )	$\lim_{x \to a} f(x) = L$
$\lim_{x \to a^-} = 5$	$\lim_{x \to a^-} f(x) = 5$
$\lim_{x \to a} f(x) = f(a)$	True only if $f$ is continuous at $a$

Notation Reading Practice 🎯

Translate the Sentence 🧮

For each English sentence, fill in the missing piece of the limit notation. Type the simplified expression — no extra spaces.

1) "As $x$ approaches $5$ from the right, $f(x)$ approaches $9$ ." $\lim_{x \to ?} f(x) = 9$ Enter the missing right side of the arrow (e.g. 5+ or -2-):

2) "As $x$ runs to positive infinity, $g(x)$ approaches $0$ ." $\lim_{x \to ?} g(x) = 0$

3) "As $x$ approaches $2$ from the left, $h(x)$ blows up to $-\infty$ ." $\lim_{x \to 2^-} h(x) = ?$ Enter , , or a number:

Match the Symbol to the Sentence 🔽

\lim_{x \to a^-} f(x) = L \;\;\text{and}\;\; \lim_{x \to a^+} f(x) = L.

In words: the limit exists ⟺ the left and right one-sided limits both exist and are equal.

If the two one-sided limits are different — or if one of them doesn't exist (oscillation, $\pm\infty$ ) — the two-sided limit does not exist (DNE).

📐 Walkthrough — A Piecewise Function

Let $f(x) = \begin{cases} 2x + 1 & x < 1 \\ 5 - x & x \ge 1 \end{cases}$

Step 1. Compute $\lim_{x \to 1^-} f(x)$ . From the left we use $2x + 1$ :

Step 2. Compute $\lim_{x \to 1^+} f(x)$ . From the right we use $5 - x$ :

Step 3. Compare. Left limit $3 \ne$ right limit $4$ , so by the existence theorem, $\boxed{\lim_{x \to 1} f(x) \;\text{does not exist.}}$

💡 The graph would show a jump discontinuity at $x = 1$ — exactly the visual signature of unequal one-sided limits.

📌 Why "Same Value" Matters

A limit is a destination. If walking toward $a$ from the left lands you at one place, and walking from the right lands you somewhere else, then there is no single destination — and so no limit.

Scenario	Left limit	Right limit	Two-sided limit
Continuous (best case)	5	5	5 (exists)
Removable hole	5	5	5 (exists; $f(a)$ may differ)
Jump	5	7	DNE
One-sided blow-up	5	$+\infty$	DNE
Two-sided blow-up	$+\infty$	$+\infty$	DNE (we say " $=\infty$ " descriptively)

🧭 The Diagnostic Procedure

When asked "Does $\lim_{x \to a} f(x)$ exist?" follow this checklist:

Try direct substitution. If $f(a)$ is defined and the function is "well-behaved" (polynomial, rational with $f(a) \ne 0/0$ , etc.), the limit equals $f(a)$ .
If you get $0/0$ , try algebraic simplification (factor, rationalize, common denominator) — Parts 5–7 cover these.
If the function is piecewise or has an absolute value, compute both one-sided limits separately and compare.
If a denominator $\to 0$ but numerator $\not\to 0$ , check the sign on each side — usually a vertical asymptote / DNE.

🎯 AP test framing: "Does the limit exist?" almost always means "do the left and right one-sided limits agree?"

Existence Concept Check 🎯

Apply the Theorem 🧮

Let $f(x) = \begin{cases} x^2 + 1 & x \le 2 \\ 3x - 1 & x > 2 \end{cases}$ .

1) $\lim_{x \to 2^-} f(x) = ?$ (whole number)

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

3) Does $\lim_{x \to 2} f(x)$ exist? Answer yes or no.

4) If yes, what is its value? (If no, type DNE.)

Diagnose the Limit 🔽

For each scenario, decide whether $\lim_{x \to a} f(x)$ exists.

Left: $\lim_{x \to 0^-} \dfrac{|x|}{x} = \dfrac{-x}{x} = -1$ .
Right: $\lim_{x \to 0^+} \dfrac{|x|}{x} = \dfrac{x}{x} = +1$ .
Conclusion: $\lim_{x \to 0} f(x)$ DNE — jump of size 2 at the origin.

💡 Visual signature: a graph that "jumps up" or "jumps down" at $a$ with a step.

2️⃣ Unbounded (vertical asymptote)

The function values grow without bound on at least one side of $a$ .

Canonical example: $f(x) = \dfrac{1}{x}$ near $x = 0$ .

Left: $\lim_{x \to 0^-} \dfrac{1}{x} = -\infty$ (large negative).

🔑 Notation note: When a limit equals $\pm\infty$ , the limit technically does not exist — but we write $\lim = \pm\infty$ as a description of the unbounded behavior.

A subtler case: $f(x) = \dfrac{1}{x^2}$ . Both sides blow up to $+\infty$ , so we say (unbounded — DNE in the strict numerical sense, but the description is meaningful since both sides agree).

3️⃣ Oscillation

The function refuses to settle on any value as $x$ approaches $a$ .

Canonical example: $f(x) = \sin\!\left(\dfrac{1}{x}\right)$ as $x \to 0$ .

As $x$ shrinks toward 0, $1/x$ races to $\infty$ , and $\sin(1/x)$ keeps oscillating between $-1$ and $+$ — infinitely many times in any tiny window around 0. There is .

Left: oscillates — DNE.
Right: oscillates — DNE.
Conclusion: $\lim_{x \to 0} \sin(1/x)$ DNE.

💡 Test for oscillation: if you can build two sequences $x_n \to a$ that yield different output sequences, the limit does not exist.

📊 Summary Table

Failure mode	Visual signature	Example	One-sided limits
Jump	Step in the graph	$\\|x\\|/x$ at 0	both exist but unequal
Unbounded	Vertical asymptote	$1/x$ at 0; $\tan x$ at $\pi/2$	one or both = $\pm\infty$
Oscillation	Infinitely wiggly	$\sin(1/x)$ at 0	one or both fail to exist

🎯 On the AP exam, you should be able to identify which failure mode is present and explain it in one sentence.

Identify the Failure Mode 🎯

Diagnose Numerically 🧮

Consider $f(x) = \dfrac{|x - 4|}{x - 4}$ .

1) $f(3.99) = ?$ (whole number with sign, e.g. -1)

2) $f(4.01) = ?$ (whole number with sign)

3) Does $\lim_{x \to 4} f(x)$ exist? Type yes or no.

4) What is the failure mode? Type jump, blowup, oscillation, or none:

Failure Mode Match 🔽

lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = L ⟺ lim_{x \to a} f (x) = L

\lim_{x \to 1^-} (2x+1) = 2(1) + 1 = 3.

lim_{x \to 1^{-}} (2 x + 1) = 2 (1) + 1 = 3.

lim_{x \to 1^{+}} (5 - x) = 5 - 1 = 4.

Right:

\lim_{x \to 0^+} \dfrac{1}{x} = +\infty

(large positive).

Conclusion: limit DNE; we say

f

has a vertical asymptote at

x = 0

\lim_{x \to 0} \frac{1}{x^2} = +\infty

lim_{x \to 0} \frac{1}{x ^{2}} = + \infty

no destination value

What is a Limit?

What is a Limit? - Complete Interactive Lesson

Part 1: The Intuitive Idea of a Limit

🎯 What Is a Limit?

Topics in This Part

What You'll Master in Part 1

💡 The Big Idea

🚗 An Everyday Analogy

📌 lim⁡x→af(x)\lim_{x\to a} f(x)limx→a​f(x) vs. f(a)f(a)f(a)

Part 2: Limit Notation

📝 Limit Notation

Topics in This Part

Part 3: When Does a Limit Exist?

✅ When Does a Limit Exist?

Topics in This Part

🔑 The Existence Theorem

Part 4: Limits That Do Not Exist

🚫 Limits That Do NOT Exist

Topics in This Part

1️⃣ Jump

Three crucial words

Three things that can happen

📌 Standard Two-Sided Limit Notation

🔑 One-Sided Limits

♾️ Limits Involving Infinity

1. Limits at infinity (end behavior of fff)

2. Limits equal to infinity (vertical asymptotes)

⚠️ Common Notation Pitfalls

📐 Walkthrough — A Piecewise Function

📌 Why "Same Value" Matters

🧭 The Diagnostic Procedure

2️⃣ Unbounded (vertical asymptote)

3️⃣ Oscillation

📊 Summary Table

📌 $\lim_{x\to a} f(x)$ vs. $f(a)$

1. Limits at infinity (end behavior of $f$ )