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Multiple Representations
of Limits

Master how to connect graphical, numerical, and algebraic representations to determine limits with confidence.

AP Calculus BC

What Is a Limit?

A limit describes the value a function approaches as the input approaches a particular point. We write:

The function does not need to equal \( L \) at \( x = c \). The limit is about the trend, not the value at the point.

Three Representations of a Limit

Graphical
Read the y-value the curve approaches as \( x \to c \) from both sides on the graph.
Numerical
Build a table of \( f(x) \) values for \( x \) approaching \( c \) from the left and right.
Algebraic
Simplify, factor, or apply limit laws to compute \( \lim_{x \to c} f(x) \) symbolically.

Why Connect Representations?

1
Verify results: Checking a limit graphically and numerically confirms your algebraic answer.
2
Handle tricky functions: Some limits are easier to see on a graph than to compute algebraically.
3
AP Exam strategy: Questions may give you a table, a graph, or an expression and ask you to determine the limit from that representation.
4
Deeper understanding: Connecting representations builds intuition about continuity, asymptotes, and behavior near critical points.

The Graphical Approach

x y 1 2 3 1 2 0
\( f(2) = 2 \)
\( \text{Limit} = 1 \)
 from left from right

The Numerical Approach

Consider \( f(x) = \dfrac{x^2 - 4}{x - 2} \). Direct substitution gives \( \frac{0}{0} \), so we build a table of 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

The Algebraic Approach

Continuing with \( f(x) = \dfrac{x^2 - 4}{x - 2} \), we factor and simplify:

1
Factor the numerator: \( x^2 - 4 = (x-2)(x+2) \)
2
Cancel common factors: \( \dfrac{(x-2)(x+2)}{x-2} = x + 2 \) for \( x \neq 2 \)
3
Substitute directly: \( \lim_{x \to 2} (x+2) = 2 + 2 = \) 4

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 \( c \).

Limit vs. Function Value

Feature Limit \( \lim_{x \to c} f(x) \) Value \( f(c) \)
What it asksWhere is \( f(x) \) heading?What is \( f(c) \)?
Depends on \( f(c) \)?No: ignores \( x = c \)Yes: requires definition at \( c \)
Can exist when \( f(c) \) is undefined?YesNo
Graphical clueFollow both branches toward \( c \)Look at the dot (filled) at \( c \)
When they agree\( f \) is continuous at \( c \)

Graphical Signals to Watch

Smooth Pass-Through
Curve passes through smoothly: limit equals the function value. Continuous at that point.
Removable Hole
Open circle with branches meeting: limit exists but \( f(c) \) is missing or different. Removable discontinuity.
Jump
Left and right branches approach different y-values: one-sided limits disagree. Two-sided limit DNE.
Vertical Asymptote
Curve shoots to \( \pm\infty \): the limit is infinite or does not exist in the finite sense.

Step-by-Step Strategy

1
Try direct substitution. If \( f(c) \) gives a real number (no \( \frac{0}{0} \) or \( \frac{k}{0} \)), that is the limit.
2
If indeterminate: simplify algebraically: factor, rationalize, or use L'Hôpital's Rule.
3
Check numerically: Build a table of values approaching \( c \) from both sides to confirm convergence.
4
Verify graphically: Sketch or examine the graph near \( x = c \). Do both branches meet at the same height?
5
State your conclusion: If all three agree, state the limit. If one-sided limits differ, the two-sided limit does not exist.

Example 1: Removable Discontinuity

Find \( \lim_{x \to 3} \dfrac{x^2 - 9}{x - 3} \) using all three representations.

x y 3 2 6
\( \text{Limit} = 6 \)

Example 2: Piecewise Function

Evaluate \( \lim_{x \to 1} f(x) \) where:

x y 1 2 3 3 4
\( f(1) = 5 \)
\( \text{Limit} = 3 \)

Example 3: Jump Discontinuity

Does \( \lim_{x \to 0} g(x) \) exist for:

x y 0 2 4
\( \text{Left limit} = 2 \)
\( \text{Right limit} = 4 \)
 JUMP

Common Mistakes

✕ Confusing Limit with Value
Thinking \( \lim_{x \to c} f(x) = f(c) \) always. The limit can differ from the function value when there is a hole or jump.
✕ Ignoring One Side
Checking only the left-hand or right-hand limit and declaring the two-sided limit exists. You must check both sides.
✕ Table Too Coarse
Using values far from \( c \) (like \( x = 1 \) and \( x = 3 \) for \( c = 2 \)). Use values like 1.99, 1.999, 2.001, 2.01 for reliable evidence.

Representation Summary

Aspect Graphical Numerical Algebraic
Tool Graph of \( f \) Table of values Symbolic manipulation
Shows one-sided limits? Yes: follow branches Yes: separate columns Must analyze each piece
Detects holes? Open vs. filled circles Undefined row at \( c \) \( \frac{0}{0} \) form
Exact answer? Approximate Approximate Exact
Best for Quick intuition Confirming trends Rigorous proof

Question 1 of 5

Given that \( \lim_{x \to 4^-} f(x) = 7 \) and \( \lim_{x \to 4^+} f(x) = 7 \), and \( f(4) = 3 \), what is \( \lim_{x \to 4} f(x) \)?

Question 2 of 5

A table shows \( f(1.9)=5.8 \), \( f(1.99)=5.98 \), \( f(1.999)=5.998 \), \( f(2.001)=6.002 \), \( f(2.01)=6.02 \), \( f(2.1)=6.2 \). What does the table suggest about \( \lim_{x \to 2} f(x) \)?

Question 3 of 5

Which algebraic technique would you use first to evaluate \( \lim_{x \to 5} \dfrac{x^2 - 25}{x - 5} \)?

Question 4 of 5

On a graph, you see a curve approaching \( y = 3 \) from the left side of \( x = 2 \), and approaching \( y = 5 \) from the right side of \( x = 2 \). What can you conclude?

Question 5 of 5

If \( \lim_{x \to c} f(x) = f(c) \), which of the following must be true?