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Estimating Limit Values
from Graphs

Learn to read graphs and determine what value a function approaches as \( x \) gets close to a point: even when the function isn't defined there.

AP Calculus BC

What Is a Limit?

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

The limit is about the trend of the outputs, not the actual function value at \( x = c \). The function might not even be defined at \( c \), yet the limit can still exist.

Why Graphs Matter for Limits

Graphs give you a visual story of a function. To estimate a limit from a graph, you trace the curve from both sides and watch where the \( y \)-values are heading.

x y c L from left from right

One-Sided Limits

Sometimes we need to examine each direction separately. These are called one-sided limits.

Left-Hand Limit
\( \displaystyle\lim_{x \to c^-} f(x) \): approach \( c \) from values less than \( c \). Start to the left of \( c \) and trace the curve toward it.
Right-Hand Limit
\( \displaystyle\lim_{x \to c^+} f(x) \): approach \( c \) from values greater than \( c \). Start to the right of \( c \) and trace the curve toward it.

When Does a Limit Exist?

1
Both one-sided limits must exist (the curve approaches a definite height from each side).
2
Both one-sided limits must be equal. If the left-hand limit is 3 and the right-hand limit is 5, the two-sided limit does not exist (DNE).
3
The function value at \( c \) does not matter. \( f(c) \) can be undefined, equal to \( L \), or something entirely different.

Limit vs. Function Value

x y 3
\(f(3) = 3.6\)
 limit = 2 2 3.6

Exists vs. Does Not Exist

Limit Exists c
\(L\)
 Limit DNE (Jump) c
\(L_1\)
\(L_2\)
Limit Exists
Left and right sides approach the same \( y \)-value. Both one-sided limits agree.
Limit DNE
A jump discontinuity: the left side approaches \( L_1 \) while the right approaches \( L_2 \) with \( L_1 \neq L_2 \).

Other Reasons a Limit May Not Exist

Unbounded Behavior
The function increases or decreases without bound near \( c \) (vertical asymptote). \( f(x) \to \pm\infty \).
Oscillation
The function oscillates infinitely fast (like \( \sin(1/x) \) near \(0\)) and never settles to one value.
Jump
Left-hand and right-hand limits are different finite values (piecewise functions, step functions).

How to Estimate a Limit from a Graph

1
Identify the target \( x \)-value. Locate \( x = c \) on the horizontal axis and draw a mental vertical line.
2
Trace from the left. Follow the curve as \( x \to c^- \). What \( y \)-value is it heading toward?
3
Trace from the right. Follow the curve as \( x \to c^+ \). What \( y \)-value is it heading toward?
4
Compare. If both sides approach the same value \( L \), then \( \lim_{x \to c} f(x) = L \). If they differ, the limit DNE.
5
Check for special cases. Does the graph shoot to \( \pm\infty \)? Oscillate? Note these as DNE.

Example 1: Removable Discontinuity

x y 2 1 3 4 3 5
\(f(2) = 5\)
 hole at y = 3

Example 2: Jump Discontinuity

x y 2 1 3 1 3
left \(\to 1\)
right \(\to 3\)

Example 3: Vertical Asymptote

x y 1 2 -1 -2
\(\to +\infty\)
\(\to -\infty\)
\(x=0\)

Estimating Limits at Infinity

You can also estimate limits as \( x \to \infty \) or \( x \to -\infty \) by looking at the end behavior of the graph: what height does the curve flatten out toward?

x y
\(y=2\)
curve \(\to 2\) as \(x \to \infty\)

Tables Support Graph Reading

Sometimes you are given a table of values near \( x = c \). The same logic applies: look at the \( y \)-values and see what they approach.

\( x \) 1.9 1.99 1.999 2 2.001 2.01 2.1
\( f(x) \) 3.61 3.96 3.996 ? 4.004 4.04 4.41

Common Mistakes to Avoid

✕ Using the Function Value
Confusing \( f(c) \) with \( \lim_{x \to c} f(x) \). The filled dot is NOT the limit: follow the trend of the curve, not the plotted point at \( x = c \).
✕ Ignoring One Side
Checking only the left-hand or right-hand approach. You must verify both sides agree before declaring the two-sided limit exists.
✕ Confusing DNE with 0
A limit of \(0\) is a valid, existing limit. "Does Not Exist" means the one-sided limits disagree or are unbounded: it does NOT mean "equals zero."

Summary of Graph Scenarios

Graph Feature Left-Hand Right-Hand Two-Sided Limit
Hole (removable) \( L \) \( L \) \( L \) (exists)
Jump discontinuity \( L_1 \) \( L_2 \neq L_1 \) DNE
Vertical asymptote (same side) \( +\infty \) \( +\infty \) DNE (unbounded)
Vertical asymptote (opposite) \( -\infty \) \( +\infty \) DNE
Continuous at \( c \) \( f(c) \) \( f(c) \) \( f(c) \) (exists)

Question 1 of 5

3 4 2

Based on the graph above, what is \( \displaystyle\lim_{x \to 3} f(x) \)?

Question 2 of 5

1 1 3

What is \( \displaystyle\lim_{x \to 1} f(x) \) for the graph shown?

Question 3 of 5

A graph shows a function that is continuous at \( x = 5 \), with a filled dot at \( (5, 7) \). The curve approaches \( y = 7 \) smoothly from both sides. What is \( \displaystyle\lim_{x \to 5} f(x) \)?

Question 4 of 5

A graph of \( g(x) \) shows a vertical asymptote at \( x = -2 \). From the left, \( g(x) \to +\infty \), and from the right, \( g(x) \to +\infty \). What is \( \displaystyle\lim_{x \to -2} g(x) \)?

Question 5 of 5

For which statement is the following always true?
"If \( \displaystyle\lim_{x \to c} f(x) = L \), then \( f(c) = L \)."