What is a Limit?

🎯 What Is a Limit?

Part 1 of 7 — The intuitive idea

Topics in This Part

🔑 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

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

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

Three crucial words

💡 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.

🚗 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 $a$?" — that's $L$.
• 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.

📌 $\lim_{x\to a} f(x)$ vs. $f(a)$

These are different questions.

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.

Concept Check 🎯

Numerical Sanity Check 🧮

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

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)

Match the Concept 🔽

Exit Check ✅

📝 Limit Notation

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

Topics in This Part

🔑 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

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

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:

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

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

🔑 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: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \;\Longleftrightarrow\; \lim_{x \to a} f(x) = L$

♾️ 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.)

⚠️ Common Notation Pitfalls

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 infinity, -infinity, or a number:

Match the Symbol to the Sentence 🔽

Exit Check ✅

✅ When Does a Limit Exist?

Part 3 of 7 — The "left = right" rule

Topics in This Part

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

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

Step 2. Compute $\lim_{x \to 1^+} f(x)$. From the right we use $5 - x$: $\lim_{x \to 1^+} (5 - x) = 5 - 1 = 4.$

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.

🧭 The Diagnostic Procedure

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

1. 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 $f(a)$.
2. If you get $0/0$, try algebraic simplification (factor, rationalize, common denominator) — Parts 5–7 cover these.
3. If the function is piecewise or has an absolute value, compute both one-sided limits separately and compare.
4. 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.

Exit Check ✅

🚫 Limits That Do NOT Exist

Part 4 of 7 — The three classic failure modes

Topics in This Part

🔑 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: $f(x) = \dfrac{|x|}{x}$.

• 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).
• Right: $\lim_{x \to 0^+} \dfrac{1}{x} = +\infty$ (large positive).
• Conclusion: limit DNE; we say $f$ has a vertical asymptote at $x = 0$.

🔑 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 $\lim_{x \to 0} \frac{1}{x^2} = +\infty$ (unbounded — DNE in the strict numerical sense, but the description $+\infty$ 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 $+1$ — infinitely many times in any tiny window around 0. There is no destination value.

• 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

🎯 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 🔽

Exit Check ✅