Estimating Limits from Graphs

📈 Estimating Limits from a Graph

Part 1 of 4 — Reading $\lim_{x \to a} f(x)$ off a picture

Topics in This Part

🔑 Why this matters: AP free-response problems regularly hand you a graph and ask "what is $\lim_{x \to 2} f(x)$?" — no formula given.

💡 The Visual Idea

A limit asks: as $x$ slides toward $a$ along the $x$-axis, what $y$-value does the graph approach?

💡 Trick of the eye: ignore the actual point above $x = a$. Look only at where the curve is heading from each side.

The limit is the $y$-value the curve is aiming for, not necessarily the dot drawn at $x = a$.

🔑 Step-by-Step: Trace Your Finger

To estimate $\lim_{x \to a} f(x)$:

1. Place your finger on the curve well to the left of $x = a$.
2. Slide your finger along the curve toward $x = a$. Watch the $y$-value.
3. Note the $y$-value the curve approaches → this is the left-hand limit $\lim_{x \to a^-} f(x)$.
4. Repeat from the right: slide from the right toward $x = a$, note the $y$-value.
5. If both arrive at the same $y$-value $L$, that's the limit. If they disagree, the limit DNE.

🔑 At step 5, the value the curve "would land on if continued" is the limit — even if there's a hole there.

📝 Easy Example

A continuous parabola $y = (x - 2)^2 + 1$ passes through $(2, 1)$ smoothly.

• Approach $x = 2$ from the left along the curve → $y$ heads to 1.
• Approach $x = 2$ from the right → $y$ heads to 1.
• Both sides agree: $\lim_{x \to 2} f(x) = 1$.

💡 For continuous functions (no breaks/holes), the limit equals the visible $y$-value at $x = a$.

Concept Check 🎯

Read the limit 🧮

A graph shows a smooth curve approaching $y = -3$ from both sides as $x \to 7$. There is an open circle at $(7, -3)$ (so $f(7)$ is undefined).

1) $\lim_{x \to 7} f(x) = ?$

2) Is $f(7)$ defined? Type yes or no.

📈 One-Sided Limits from a Graph

Part 2 of 4 — Left arm, right arm

Topics in This Part

🔑 Why this matters: AP graphs love to show piecewise functions where the two sides do different things.

👈👉 Left vs. Right Visually

Two notations on the same graph:

• $\lim_{x \to a^-} f(x)$: trace from the left toward $a$. Read the $y$-value the curve heads for.
• $\lim_{x \to a^+} f(x)$: trace from the right toward $a$. Read the $y$-value the curve heads for.

💡 Tip: the superscript $-$ means "from values less than $a$" (left). The superscript $+$ means "from values greater than $a$" (right).

✅ The Existence Theorem — Visually

Two-sided limit exists $\iff$ both arms of the graph aim at the same $y$-value at $x = a$.

The value $f(a)$ — drawn as a closed dot, open circle, or missing entirely — does NOT affect the limit.

📖 Worked Reading

A piecewise graph $f$ near $x = 1$ shows:

• Left arm aiming at $y = 4$ (open circle at $(1, 4)$).
• Right arm aiming at $y = 2$ (closed dot at $(1, 2)$).

⚠️ Don't confuse "$f(1)$ exists" with "the limit at 1 exists." They are independent.

Read the One-Sided Limits 🎯

Picture Match 🔽

🚫 When the Graph Shows DNE

Part 3 of 4 — Visual signatures of failure

Topics in This Part

🔑 Why this matters: On a graph-only AP problem, you must say not just "DNE" but why.

1️⃣ Visual Signature: Jump

A step in the curve. The two arms aim at different finite $y$-values.

Think of a staircase, or the floor function $\lfloor x \rfloor$:

• Left arm at $x = 2$ aims at $y = 1$.
• Right arm at $x = 2$ aims at $y = 2$.
• Limit DNE — jump discontinuity.

💡 Drawing tell: a vertical "step" with often a closed dot on one side and an open circle on the other.

2️⃣ Visual Signature: Vertical Asymptote

The curve shoots off to $\pm\infty$ as $x$ approaches $a$.

Think $1/x$ at $x = 0$:

• Left arm dives to $-\infty$.
• Right arm shoots to $+\infty$.
• Limit DNE — unbounded blow-up; vertical asymptote at $x = 0$.

A subtler case is $1/x^2$ at 0: both arms shoot to $+\infty$. We describe the behavior as $\lim = +\infty$, but the limit does not exist as a finite number.

💡 Drawing tell: a dashed vertical line, with the curve hugging it asymptotically.

3️⃣ Visual Signature: Oscillation

The curve wiggles infinitely fast near $a$ — never settling on any value.

Think $\sin(1/x)$ at $x = 0$: as you slide toward 0, the graph crashes through $-1$ and $+1$ infinitely many times in any tiny window.

• Limit DNE — oscillation.

💡 Drawing tell: a "fuzzy" or shaded vertical band near $a$ where the curve cycles densely between two horizontal levels.

Diagnose from the Picture 🎯

Match the Picture to the Failure Mode 🔽

🔵🟡 Holes, Open/Closed Dots, and $f(a)$ vs. $\lim$

Part 4 of 4 — Decoding dot conventions

Topics in This Part

🔑 Why this matters: Misreading dots is the #1 source of lost points on graph problems.

🔑 Open vs. Closed Dot Convention

💡 The limit cares only about where the curve is heading, never about which dot is filled.

📊 Limit vs. Value: A Side-by-Side

For one graph at $x = 3$:

🔑 The four cases above show why we separate "the limit at $a$" from "the value $f(a)$" — they are completely different questions.

📝 Common AP Graph Setups

You'll often see piecewise graphs with deliberate "trap" features at one $x$-value:

• A removable hole (open circle, no replacement dot).
• A limit that exists but $f(a)$ is "moved" to a different $y$ via a closed dot elsewhere.
• A jump where one side has a closed dot and the other an open circle.

💡 Strategy: answer questions in this order: (1) left limit, (2) right limit, (3) two-sided limit, (4) $f(a)$. Doing them separately prevents conflating limit and value.

Decode the Dots 🎯

Final Reading 🧮

A graph at $x = 5$ shows: left arm $\to 3$, right arm $\to 3$, an OPEN circle at $(5, 3)$, no other dot drawn.

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

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

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

4) $f(5) = ?$ (use undefined if no value is defined)

Exit Check ✅