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AP Calculus AB Practice Quiz: Connecting Multiple Representations of Limits

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

The expression lim (x→c) f(x) = L is an example of which type of representation for a limit?

All Questions (7)

The expression lim (x→c) f(x) = L is an example of which type of representation for a limit?

A) Graphical

B) Numerical

C) Analytic

D) Tabular

Correct Answer: C

This expression uses mathematical symbols and notation to define the limit, which is known as an analytic representation. A graphical representation would be a graph, and a numerical representation would be a table of values.

A graph of a function g(x) shows that as the x-values get arbitrarily close to 4 from both sides, the corresponding y-values get arbitrarily close to 10. Which of the following analytic notations correctly interprets this graphical behavior?

A) g(4) = 10

B) lim (x→10) g(x) = 4

C) lim (x→4) g(x) = 10

D) lim (x→∞) g(x) = 10

Correct Answer: C

The concept of x-values approaching a number (4) and causing y-values to approach another number (10) is the definition of a limit. The correct analytic notation for this is lim (x→4) g(x) = 10. The expression g(4) = 10 would describe the value of the function exactly at x=4, which is not necessarily the same as the limit.

A student creates the following table of values for a function f(x): x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 --|---|---|---|---|---|--- f(x) | -0.8 | -0.98 | -0.998 | -1.002 | -1.02 | -1.2 Based on this numerical representation, which analytic expression is the most likely interpretation?

A) lim (x→2) f(x) = -1

B) lim (x→-1) f(x) = 2

C) f(2) = -1

D) lim (x→-1) f(x) = ∞

Correct Answer: A

The table provides a numerical representation of the limit. It shows that as the x-values get closer to 2 from both the left (1.999) and the right (2.001), the f(x) values get closer to -1. This supports the analytic expression lim (x→2) f(x) = -1.

Which of the following scenarios best describes a numerical approach to representing the limit of a function f(x) as x approaches a?

A) Sketching the curve of y = f(x) and observing its height as it nears the vertical line x = a.

B) Using algebraic rules and theorems to simplify the expression of f(x) and substitute the value of a.

C) Creating a table of f(x) values for x-values that are progressively closer to a from both the left and right sides.

D) Writing the formal epsilon-delta definition for the limit of the function at x = a.

Correct Answer: C

A numerical representation of a limit involves using numbers (typically in a table) to see how the function's output behaves as the input approaches a certain value. Option A is graphical, and options B and D are analytical.

The analytic statement lim (x→-3) f(x) = 5 is given. Which of the following must be true about the graphical representation of the function f(x)?

A) The graph must have a solid, filled-in point at the coordinates (-3, 5).

B) The graph must have a vertical asymptote at x = -3.

C) The y-values of the graph must approach 5 as the x-values approach -3 from both sides.

D) The graph must have a horizontal asymptote at y = 5.

Correct Answer: C

The analytic notation for a limit describes the behavior of the function as it gets close to a point, not necessarily the value at the point itself. Graphically, this means the curve of the function gets arbitrarily close to a y-value of 5 as the x-value gets arbitrarily close to -3. The function could have a hole, a jump, or be continuous at x=-3.

If a limit is expressed analytically as lim (x→0) h(x) = 2, what would one expect to see in a numerical representation (table of values) for this limit?

A) The value of h(0) is exactly 2.

B) For x-values very close to 0, like -0.001 and 0.001, the corresponding h(x) values will be very close to 2.

C) For all x-values in the table, the h(x) value is 2.

D) For x-values very close to 2, the corresponding h(x) values will be very close to 0.

Correct Answer: B

An analytic limit expression describes the function's approaching behavior. A numerical representation would show this by listing function values (h(x)) for input values (x) that are very near the target. As x gets closer to 0, h(x) should get closer to 2. The table does not need to include, nor does the limit define, the value of h(0) itself.

A function is represented graphically by a curve that has a hole at the point (c, L) but is otherwise continuous. Which statement correctly connects the different representations of this function's behavior at x = c?

A) Analytically, lim (x→c) f(x) = L, but f(c) is undefined. Numerically, a table would show f(x) values approaching L as x approaches c.

B) Analytically, lim (x→c) f(x) does not exist. Numerically, a table would show f(x) values approaching different numbers from the left and right of c.

C) Analytically, f(c) = L. Graphically, the point (c, L) is a solid dot on the curve.

D) Analytically, lim (x→c) f(x) = L and f(c) = L. Numerically, the table would show f(x) values approaching L and include the point (c, L).

Correct Answer: A

A hole at (c, L) means the function approaches L as x approaches c, so the limit exists and is L. This is the analytic expression lim (x→c) f(x) = L. However, the 'hole' itself means the function is undefined at that exact point, so f(c) is undefined. A numerical table would reflect the limit's existence by showing f(x) values getting closer to L as x gets closer to c.