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Connecting Multiple
Representations of Limits

Master the skill of evaluating limits using graphs, tables, and algebraic expressions and learn to connect all three seamlessly.

AP Calculus AB

What Is a Limit?

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

The limit may exist even if \( f(c) \) is undefined or different from \( L \). What matters is the behavior near \( c \), not at \( c \) itself.

Three Representations of Limits

Graphical
Read the y-value the curve approaches as \( x \to c \) from both sides on a graph.
Numerical (Table)
Choose \( x \)-values increasingly close to \( c \) and observe the pattern in \( f(x) \).
Algebraic
Simplify the expression analytically using factoring, conjugates, or known identities.

Why Connect Representations?

1
Verification: If the graph says one thing and algebra says another, something is wrong. Cross-checking catches errors.
2
Completeness: Some AP problems give you a table and ask for a graphical interpretation, or vice versa.
3
Deeper understanding: Knowing why algebraic cancellation works is visible in the graph as a removable discontinuity (a hole).

The Core Principle

The two-sided limit exists only when both the left-hand limit and the right-hand limit exist and are equal. Each representation reveals this:

Graph
The curve approaches the same height from both directions.
Table
Values from the left (\(x < c\)) and right (\(x > c\)) both converge to \( L \).

Reading Limits from a Graph

x y 2 3 5

\(f(2) = 5\)

\(\text{Limit} = 3\)

 from left from right

Reading Limits from a Table

Consider \( f(x) = \dfrac{x^2 - 4}{x - 2} \). We cannot plug in \( x = 2 \) directly, but we can examine nearby values:

\( x \) (from left) \( f(x) \) \( x \) (from right) \( f(x) \)
1.93.92.14.1
1.993.992.014.01
1.9993.9992.0014.001
1.99993.99992.00014.0001

Evaluating Limits Algebraically

Using the same function \( f(x) = \dfrac{x^2 - 4}{x - 2} \), we simplify:

After cancellation, we can substitute directly:

This confirms the table result and what we would see on a graph: a line \( y = x + 2 \) with a hole at \( x = 2 \).

Limit vs. Function Value

Property \(\lim_{x \to c} f(x)\) \(f(c)\)
What it asksWhat does \( f \) approach near \( c \)?What is \( f \) exactly at \( c \)?
Requires \( f(c) \) defined?NoYes
Uses behavior at \( c \)?No: only near \( c \)Yes: exactly at \( c \)
Can they differ?Yes: removable discontinuity

One-Sided Limits

\(\lim_{x \to c^-} f(x)\)
The left-hand limit: approach \( c \) from values less than \( c \). On a graph, trace the curve from the left.
\(\lim_{x \to c^+} f(x)\)
The right-hand limit: approach \( c \) from values greater than \( c \). On a graph, trace the curve from the right.

Step-by-Step Strategy

1
Try direct substitution. Plug \( c \) into \( f(x) \). If you get a real number, that is the limit.
2
If you get \(\frac{0}{0}\): Factor, rationalize, or simplify to cancel the common factor causing the issue.
3
Build a table. Choose values approaching \( c \) from both sides. Confirm the outputs converge.
4
Check the graph. Verify the curve approaches the same height from the left and right.
5
State your conclusion. Confirm all three representations agree.

Example 1: Removable Discontinuity

Evaluate \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\).

\( x \) 2.9 2.99 2.999 3.001 3.01 3.1
\( f(x) \) 5.9 5.99 5.999 6.001 6.01 6.1

Example 2: Piecewise Function

Let \( g(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases}\). Find \(\lim_{x \to 1} g(x)\).

x y 1 2 3 1 2 3

\(y = x^2\)

\(y = 2x - 1\)

Both \(\to 1\)

Example 3: When the Limit Does Not Exist

Let \( h(x) = \begin{cases} 2 & \text{if } x < 0 \\ 5 & \text{if } x \geq 0 \end{cases}\). Find \(\lim_{x \to 0} h(x)\).

x y 2 5

\(y = 2\)

\(y = 5\)

 Jump!

Common Mistakes

✕ Confusing \( f(c) \) with the Limit
The limit depends on behavior near \( c \), not at \( c \). A filled dot at a different height does not change the limit.
✕ Checking Only One Side
You must verify both \(\lim_{x \to c^-}\) and \(\lim_{x \to c^+}\). Ignoring one side can lead you to claim a limit exists when it doesn't.
✕ Plugging Into \(\frac{0}{0}\)
Getting \(\frac{0}{0}\) is not the answer. It means the limit is indeterminate and you must simplify further before evaluating.

Representation Summary

Representation How to Find the Limit Key Sign the Limit DNE
Graphical Trace the curve toward \( x = c \) from both sides; read the \( y \)-value Curve approaches different heights from left and right
Numerical Build a table with \( x \)-values approaching \( c \); look for convergence Left-side values converge to a different number than right-side values
Algebraic Simplify (factor, rationalize, etc.) then substitute \( c \) After simplification, left and right expressions give different values

Question 1 of 5

A table shows that as \( x \to 3^- \), \( f(x) \to 7 \), and as \( x \to 3^+ \), \( f(x) \to 7 \). However, \( f(3) = 10 \). What is \(\lim_{x \to 3} f(x)\)?

Question 2 of 5

Evaluate \(\displaystyle\lim_{x \to -1} \frac{x^2 - 1}{x + 1}\) algebraically.

Question 3 of 5

On a graph, the curve approaches \( y = 4 \) from the left as \( x \to 2 \), and approaches \( y = 4 \) from the right as \( x \to 2 \). There is an open circle at \( (2, 4) \) and a filled dot at \( (2, 6) \). What is \(\lim_{x \to 2} f(x)\)?

Question 4 of 5

For \( g(x) = \begin{cases} 3x + 1 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \geq 2 \end{cases}\), what is \(\lim_{x \to 2} g(x)\)?

Question 5 of 5

Direct substitution of \( x = 4 \) into \(\dfrac{x^2 - 16}{x - 4}\) gives \(\dfrac{0}{0}\). What should you do next?