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Estimating Limits
from Graphs

Learn to read graphs like a calculus pro: determine what value \( f(x) \) approaches as \( x \) gets close to a target, even when the function has holes, jumps, or asymptotes.

AP Calculus AB

What Is a Limit?

A limit describes the value that \( f(x) \) gets closer and closer to as \( x \) approaches some number \( c \). We write this as:

This means: as \( x \) gets arbitrarily close to \( c \) (from both sides), the output \( f(x) \) gets arbitrarily close to \( L \). The function does not need to equal \( L \) at \( x = c \).

Approaching vs Arriving

x y

\( c \)

\( L \) (limit)

\( f(c) \) (actual)

 \u2192 \u2190

One-Sided Limits

Left-Hand Limit
\( \lim_{x \to c^-} f(x) \): the value \( f(x) \) approaches as \( x \) comes from values less than \( c \).
Right-Hand Limit
\( \lim_{x \to c^+} f(x) \): the value \( f(x) \) approaches as \( x \) comes from values greater than \( c \).

When Does the Limit Exist?

Limit Exists Both sides agree Limit Does Not Exist Jump: sides disagree

Reading the Graph for Limits

x y 1 2 3 2 3 1

\( L = 3 \)

\( f(2) = 1 \)

Limit vs Function Value

Limit \( \lim_{x \to c} f(x) \) Function Value \( f(c) \)
What it asks Where is \( f(x) \) heading? Where does \( f(x) \) land?
Requires \( f(c) \) to exist? No Yes
Graph clue Trace the curve toward \( x = c \) Look at the dot at \( x = c \)
Affected by holes? No Yes

Three Scenarios on a Graph

Removable Hole
Open circle at \( (c, L) \). Both sides approach \( L \). Limit is \( L \).
Jump Discontinuity
Left-hand and right-hand limits exist but differ. Two-sided limit DNE (Does Not Exist).
Vertical Asymptote
Curve shoots to \( \pm\infty \). Limit is unbounded and DNE (in the finite sense).

How to Estimate a Limit

1
Locate \( x = c \) on the horizontal axis and draw a mental vertical line.
2
Trace the curve from the left (\( x \to c^- \)). Note the \( y \)-value the curve approaches.
3
Trace the curve from the right (\( x \to c^+ \)). Note the \( y \)-value the curve approaches.
4
Compare: If both sides approach the same \( y \)-value \( L \), then \( \lim_{x \to c} f(x) = L \). If they differ, the limit does not exist.
5
Ignore the dot (or lack of one) at \( x = c \). The filled/open circle there tells you about \( f(c) \), not the limit.

Example 1: Removable Hole

x y 1 2 3 2 4 3

\( L = 3 \)

 \u2192 \u2190

Example 2: Jump Discontinuity

x y 1 2 4

Left: 2

Right: 4

\( x = 1 \)

Example 3: Vertical Asymptote

x y 0 \u2191 \u2191

Asymptote: \( x = 0 \)

Both sides: \( +\infty \)

Infinite Limits vs Limits at Infinity

Infinite Limit
\( \lim_{x \to c} f(x) = \pm\infty \): \( x \) approaches a finite number \( c \), but the output blows up. Vertical asymptote at \( x = c \).
Limit at Infinity
\( \lim_{x \to \infty} f(x) = L \): \( x \) itself grows without bound, and the output settles toward a finite \( L \). Horizontal asymptote at \( y = L \).

Graphs + Tables Together

When a graph is hard to read precisely, a table of values can help you estimate the limit by picking \( x \)-values that get closer to \( c \) from both sides:

\( x \) 1.9 1.99 2 2.01 2.1
\( f(x) \) 4.61 4.96 ? 5.04 5.41

A few readings leave the intervals unseen

A probe touches a plain metal object and connects to a blank logger beside a notebook and separated blank reading cards.Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.

A few readings leave the intervals unseen

Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.

Common Mistakes

✕ Evaluating at the Point
Plugging in \( x = c \) directly. The limit asks what happens near \( c \), not at \( c \).
✕ Ignoring One Side
Checking only the left or right side. You must confirm both one-sided limits agree.
✕ Confusing Dot Types
Treating an open circle as the limit value. Open circles show the excluded value: the limit is where the curve heads.

Summary Cheat Sheet

Graph Feature Limit Outcome How to Spot It
Smooth passage Limit = \( f(c) \) Curve passes through the point
Removable hole Limit = \( L \) (may be \( \neq f(c) \)) Open circle; both sides agree
Jump Limit DNE Curve "breaks"; sides disagree
Vertical asymptote Limit DNE (unbounded, \( \pm\infty \)) Curve shoots to \( \pm\infty \)
Oscillation Limit DNE Curve vibrates infinitely fast

Question 1 of 5

A graph shows a curve approaching \( y = 4 \) from both sides as \( x \to 3 \), with an open circle at \( (3, 4) \) and a filled dot at \( (3, 1) \). What is \( \lim_{x \to 3} f(x) \)?

Question 2 of 5

On a graph, \( \lim_{x \to 2^-} f(x) = 5 \) and \( \lim_{x \to 2^+} f(x) = 5 \). Which statement must be true?

Question 3 of 5

A graph has a vertical asymptote at \( x = -1 \). From the left, the curve goes to \( +\infty \), and from the right, the curve goes to \( -\infty \). What is \( \lim_{x \to -1} f(x) \)?

Question 4 of 5

A table shows: as \( x \to 5^- \), \( f(x) \) values are 7.9, 7.99, 7.999. As \( x \to 5^+ \), \( f(x) \) values are 8.001, 8.01, 8.1. What is the best estimate for \( \lim_{x \to 5} f(x) \)?

Question 5 of 5

On a graph, the curve from the left approaches \( y = 3 \) as \( x \to 1 \), and the curve from the right approaches \( y = 6 \) as \( x \to 1 \). There is a filled dot at \( (1, 3) \). What is \( \lim_{x \to 1^+} f(x) \)?