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AP Calculus AB Practice Quiz: Defining Limits and Using Limit Notation

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

Test your understanding with short quizzes. This quiz has 13 questions to check your progress.

Question 1 of 13

Which of the following correctly uses analytic notation to represent the statement: "The limit of the function f(x) as x approaches 3 is 9"?

All Questions (13)

Which of the following correctly uses analytic notation to represent the statement: "The limit of the function f(x) as x approaches 3 is 9"?

A) $\lim_{x\to 3}f(x)=9$

B) $f(x) \to 3$ as $x=9$

C) $\lim_{x\to 9}f(x)=3$

D) $f(3)=9$

Correct Answer: A

The standard notation for a limit is $\lim_{x\to c}f(x)=R$, where c is the value x approaches and R is the limit. In this statement, x approaches 3 (so c=3) and the limit is 9 (so R=9). Option A correctly represents this.

How is the expression $\lim_{x\to a}g(x)=L$ best interpreted?

A) The value of the function g(x) at x=a is equal to L.

B) As x gets sufficiently close to a, the value of g(x) gets arbitrarily close to L.

C) The function g(x) is equal to L for all values of x near a.

D) The limit of x is L as g(x) approaches a.

Correct Answer: B

This phrasing directly matches the definition of a limit provided in the content, which states that f(x) can be made arbitrarily close to the limit R (or L in this case) by taking x sufficiently close to c (or a in this case), but not equal to c.

If the values of a function h(t) can be made arbitrarily close to -5 by taking t sufficiently close to 0, but not equal to 0, which of the following statements is the correct analytical representation?

A) $h(0) = -5$

B) $\lim_{t\to 0}h(t)=-5$

C) $\lim_{t\to -5}h(t)=0$

D) $h(t) \to 0$ as $t \to -5$

Correct Answer: B

The description provided is the verbal definition of a limit. The notation $\lim_{t\to c}h(t)=R$ represents that h(t) approaches R as t approaches c. In this case, c=0 and R=-5.

The existence and value of $\lim_{x\to c}f(x)$ depend on the values of f(x) for x near c. What does the value of the limit NOT depend on?

A) The values of f(x) for x > c.

B) The values of f(x) for x < c.

C) The value of f(c).

D) The values of f(x) being arbitrarily close to the limit.

Correct Answer: C

The definition of a limit specifies that x approaches c "but not equal to c". Therefore, the value of the function *at* c, which is f(c), does not affect the value of the limit as x approaches c.

According to the provided content, which of the following is a way that a limit can be expressed?

A) Algebraically

B) Numerically

C) Hypothetically

D) Abstractly

Correct Answer: B

The content explicitly states that "A limit can be expressed in multiple ways, including graphically, numerically, and analytically." Of the choices provided, only "Numerically" is listed.

Given that $\lim_{x\to -1}p(x)=4$, which of the following statements must be true?

A) p(-1) exists.

B) p(-1) = 4.

C) p(x) can be made arbitrarily close to 4 by taking x sufficiently close to -1.

D) p(x) is defined for all x in an interval containing -1.

Correct Answer: C

The statement $\lim_{x\to -1}p(x)=4$ is a statement about the behavior of the function *near* x=-1, not *at* x=-1. It directly means that the function's values approach 4 as x approaches -1. The function does not need to be defined at -1, nor does its value at -1 (if it exists) need to be 4.

In the standard limit notation $\lim_{x\to c}f(x)=R$, what does the symbol R represent?

A) The value that x is approaching.

B) The function being evaluated.

C) The real number that f(x) is approaching.

D) The value of the function at x=c.

Correct Answer: C

In the notation $\lim_{x\to c}f(x)=R$, R is the real number that the function values, f(x), get arbitrarily close to as x gets sufficiently close to c.

The statement "$\lim_{x\to 2}f(x)$ exists and is a real number" implies that:

A) f(2) is a real number.

B) The function f(x) is continuous at x=2.

C) As x gets closer to 2 from both sides, f(x) approaches a single real number.

D) f(x) must be a simple algebraic function.

Correct Answer: C

The existence of a limit at a point means that as the input x approaches that point, the output f(x) approaches a single, finite value. This is the core concept of a limit existing. The value of f(2) itself is not relevant to the existence of the limit.

The provided text states that a limit can be expressed analytically, graphically, and which other way?

A) Verbally

B) Numerically

C) Symbolically

D) Theoretically

Correct Answer: B

The content states, "A limit can be expressed in multiple ways, including graphically, numerically, and analytically."

A critical aspect of the definition of $\lim_{x\to c}f(x)=R$ is that f(x) can be made "arbitrarily close to R". What does this phrase signify?

A) f(x) must eventually equal R.

B) We can make the distance between f(x) and R smaller than any chosen positive number.

C) f(x) is always a rational number close to R.

D) The graph of f(x) must be a straight line that passes through the point (c, R).

Correct Answer: B

"Arbitrarily close" means that for any small positive distance you can choose, we can guarantee that the distance between f(x) and the limit R is less than that chosen distance, provided x is sufficiently close to c. This is the essence of the formal definition of a limit.

Let g be a function such that as x gets sufficiently close to 10, the values of g(x) get arbitrarily close to 25. It is also known that g(10) = -5. Based on this information, what is the value of $\lim_{x\to 10}g(x)$?

A) 25

B) -5

C) 10

D) The limit does not exist.

Correct Answer: A

The limit of a function as x approaches c is determined by the function's behavior *near* c, not *at* c. The information that g(x) approaches 25 as x approaches 10 defines the limit. The value g(10)=-5 is irrelevant to the value of the limit.

Which expression correctly represents the limit of a function k(z) as z approaches a constant b, where the limit is the real number M?

A) $\lim_{z\to b}k(z)=M$

B) $\lim_{M\to b}k(z)=z$

C) $\lim_{k(z)\to M}z=b$

D) $k(b) \to M$

Correct Answer: A

The standard notation is $\lim_{\\text{variable}\to \\text{constant}} \\text{function} = \\text{limit value}$. In this case, the variable is z, the constant it approaches is b, the function is k(z), and the limit value is M. Option A correctly arranges these components.

If a student writes $\lim_{x\to 5}f(x)=R$, what must be true about the number R, assuming the limit exists?

A) R must be equal to f(5).

B) R must be an integer.

C) R must be a real number.

D) R must be greater than 0.

Correct Answer: C

The provided content states, "...the limit of f(x) as x approaches c is a real number R ... If the limit exists and is a real number, then the common notation is $\lim_{x\to c}f(x)=R$." This directly indicates that for this notation to be used for an existing limit, R must be a real number.