🎯⭐ INTERACTIVE LESSON

Estimating Limits from Graphs

Learn step-by-step with interactive practice!

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Estimating Limits from Graphs - Complete Interactive Lesson

Part 1: Reading $\lim$ Off a Graph

📈 Estimating Limits from a Graph

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

Topics in This Part

Section
The Visual Idea
🔑 Step-by-Step: Trace Your Finger
Easy Example

🔑 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 is heading from each side.

🔑 Step-by-Step: Trace Your Finger

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

Place your finger on the curve well to the left of $x = a$ .
Slide your finger along the curve toward $x =$ . Watch the -value.

📝 Easy Example

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

Approach from the left along the curve → heads to 1.

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).

Part 2: One-Sided Limits Visually

📈 One-Sided Limits from a Graph

Part 2 of 4 — Left arm, right arm

Topics in This Part

Section
Left vs. Right Visually
The Existence Theorem (visual form)
Worked Reading

🔑 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 toward . Read the -value the curve heads for.

Part 3: Visual Signatures of DNE

🚫 When the Graph Shows DNE

Part 3 of 4 — Visual signatures of failure

Topics in This Part

Section
Visual Signature: Jump
Visual Signature: Vertical Asymptote
Visual Signature: Oscillation

🔑 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 .

Part 4: Open/Closed Dots vs Limit vs Value

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

Part 4 of 4 — Decoding dot conventions

Topics in This Part

Section
🔑 Open vs. Closed Dot Convention
Limit vs. Value: A Side-by-Side
Common AP Graph Setups

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

🔑 Open vs. Closed Dot Convention

Symbol on the graph at $(a, y)$

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

Note the

y

-value the curve approaches → this is the left-hand limit

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

Repeat from the right: slide from the right toward

x = a

, note the

y

-value.

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.

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 $x = a$ .

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

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

\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 $x = a$ .

Picture at $x = a$	What it means
Both arms meet at the same $y$	$\lim_{x \to a} f(x) =$ that $y$ -value
Arms aim at different -values (graph "jumps")

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 $(1, 2)$ ).

Question	Answer
$\lim_{x \to 1^-} f(x)$	$4$

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

Read the One-Sided Limits 🎯

Picture Match 🔽

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.

Failure mode	Visual signature
Jump	Step in graph
Vertical asymptote	Curve shoots to $\pm\infty$
Oscillation	Fuzzy/wiggly band

Diagnose from the Picture 🎯

Match the Picture to the Failure Mode 🔽

💡 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 $x = 3$ :

Picture detail	$\lim_{x \to 3} f(x)$	$f(3)$
Both arms meet at 5; closed dot at $(3, 5)$	$5$	$5$ — continuous!
Both arms meet at 5; open circle at $(3, 5)$ , no other dot	$5$

🔑 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)

\lim_{x \to 1^+} f(x)

lim_{x \to 1^{+}} f (x)

Closed (filled) dot	$f(a) = y$ — the function takes this value
Open (hollow) circle	$f(a) \ne y$ — the curve approaches $y$ but does NOT take it there
No mark, just the curve continuing	Read $f(a)$ from the curve itself

Estimating Limits from Graphs

Estimating Limits from Graphs - Complete Interactive Lesson

Part 1: Reading $\lim$ Off a Graph

📈 Estimating Limits from a Graph

Topics in This Part

💡 The Visual Idea

🔑 Step-by-Step: Trace Your Finger

📝 Easy Example

Part 2: One-Sided Limits Visually

📈 One-Sided Limits from a Graph

Topics in This Part

👈👉 Left vs. Right Visually

Part 3: Visual Signatures of DNE

🚫 When the Graph Shows DNE

Topics in This Part

1️⃣ Visual Signature: Jump

Part 4: Open/Closed Dots vs Limit vs Value

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

Topics in This Part

🔑 Open vs. Closed Dot Convention

✅ The Existence Theorem — Visually

📖 Worked Reading

2️⃣ Visual Signature: Vertical Asymptote

3️⃣ Visual Signature: Oscillation

📊 Limit vs. Value: A Side-by-Side

📝 Common AP Graph Setups

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