AP Calculus AB Quiz: Defining Limits And Using Limit Notation

What this quiz covers

This quiz focuses on Defining Limits And Using Limit Notation, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus AB.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $\displaystyle \lim_{x\to 3^-} p(x)=-2$ (correct answer)
2. $\displaystyle \lim_{x\to 3^+} p(x)=-2$
3. $\displaystyle p(3)=-2$
4. $\displaystyle \lim_{x\to -2} p(x)=3$
5. $\displaystyle \lim_{x\to 3} p(3)=-2$

Explanation: Limit notation highlights approaching values, and the graph shows p(x) nearing -2 from the left as x approaches 3. Hence, \lim_{x\to 3^-} p(x)=-2 is the precise left-hand limit expression. This is valid for capturing behavior from x less than 3. A frequent symbolic mistake is \lim_{x\to 3} p(3)=-2, incorrectly placing the evaluation inside. Using p(3)=-2 confuses limit with function value. Specify one-sided when sides differ. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 2

1. $\displaystyle \lim_{x\to 2} r(x)=-3$ (correct answer)
2. $\displaystyle \lim_{x\to -3} r(x)=2$
3. $\displaystyle \lim_{x\to 2} r(x)=r(2)$
4. $\displaystyle \lim_{x\to 2^-} r(x)=3$
5. $\displaystyle \lim_{x\to 2^+} r(x)=\text{DNE}$

Explanation: An open circle on a graph signifies the limit value where the function is not defined or differs, but the curve's approach determines the limit. The graph approaches -3 from both sides at x=2 with an open circle, so \lim_{x \to 2} r(x) = -3 is correct. This notation is valid for the bilateral approach shown. A common symbolic error is assuming DNE due to the hole, ignoring the approaching behavior. Another mistake is using one-sided notation unnecessarily, like \lim_{x \to 2^-} r(x) = 3, or equating to r(2). Verify graph symmetry near the point. Transferable notation checklist: 1. Write as \lim_{x \to a} f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

Question 3

1. $\lim_{x\to3^+} g(x)=1$
2. $\lim_{x\to3} g(x)=5$
3. $\lim_{x\to3^-} g(x)=4$
4. $\lim_{x\to3^+} g(x)=4$ (correct answer)
5. $\lim_{x\to3} g(3)=5$

Explanation: The function $g$ has a jump discontinuity at $x=3$: it equals 1 for $x<3$, jumps to 5 at $x=3$, then equals 4 for $x>3$. When approaching from the right (values greater than 3), we're in the region where $g(x)=4$, so the right-hand limit is $\lim_{x\to 3^+} g(x) = 4$. The superscript plus sign indicates we only consider values approaching from the right. A common mistake is thinking the limit must involve the function value at the point—but $g(3)=5$ is irrelevant to the right-hand limit. Another error is writing $\lim_{x\to 3} g(3)$, which incorrectly substitutes the value into the limit notation. Notation checklist: (1) $\lim$ symbol, (2) $x\to 3^+$ for right-hand approach, (3) $g(x)$ not $g(3)$, (4) equals 4.

Question 4

1. $\displaystyle \lim_{x\to 0} s(x)=0$
2. $\displaystyle \lim_{x\to 0^-} s(x)=1$
3. $\displaystyle \lim_{x\to 0^+} s(x)=-1$
4. $\displaystyle \lim_{x\to 0} s(x)=\text{DNE}$ (correct answer)
5. $\displaystyle \lim_{s(x)\to 0} x=1$

Explanation: If left and right limits differ, the two-sided limit does not exist, denoted as DNE in notation. For s(x) = |x|/x, it approaches 1 from the right and -1 from the left as x nears 0, so \lim_{x \to 0} s(x) = DNE is accurate. This is valid when sides disagree, despite s(0) = 0. A common error is claiming a limit value like 0, confusing with the function at 0. Another symbolic mistake is reversing, such as \lim_{s(x) \to 0} x = 1, or using one-sided without specifying DNE for two-sided. Always check both directions for existence. Transferable notation checklist: 1. Write as \lim_{x \to a} f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

Question 5

1. $\displaystyle \lim_{x\to -1^+} H(x)=-2$ (correct answer)
2. $\displaystyle \lim_{x\to -1} H(x)=-2$
3. $\displaystyle H(-1)=-2$
4. $\displaystyle \lim_{x\to -1^-} H(x)=-2$
5. $\displaystyle \lim_{x\to -2} H(x)=-1$

Explanation: Graph H(x) to -2 from right at -1. \lim_{x\to -1^+} H(x)=-2 correct. Valid right. Error: \lim_{x\to -1} H(x)=-2. H(-1)=-2 value. Specify. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 6

1. $\displaystyle a(2)=-4$
2. $\displaystyle \lim_{x\to 2} a(x)=-4$ (correct answer)
3. $\displaystyle \lim_{x\to 2^+} a(x)=4$
4. $\displaystyle \lim_{x\to -4} a(x)=2$
5. $\displaystyle \lim_{x\to 2} a(2)=-4$

Explanation: Limits capture behavior near points, graph showing a(x) to -4 at x=2 both sides. \lim_{x\to 2} a(x)=-4 matches. Valid ignoring a(2). Error: \lim_{x\to 2} a(2)=-4. a(2)=-4 is value. Distinguish. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 7

1. $\displaystyle \lim_{x\to 1} t(x)=6$ (correct answer)
2. $\displaystyle t(1)=6$
3. $\displaystyle \lim_{x\to 6} t(x)=1$
4. $\displaystyle \lim_{x\to 1^-} t(x)=5$
5. $\displaystyle \lim_{x\to 1} t(1)=6$

Explanation: Limit notation conveys approaching values, as the table shows t(x) nearing 6 as x approaches 1. Hence, $\lim_{x\to 1} t(x)=6$ is appropriate for this behavior. This is valid regardless of $t(1)$. An error is $\lim_{x\to 1} t(1)=6$, misplacing evaluation. $t(1)=6$ is function value, not limit. Distinguish clearly. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with $x \to a$. 3. Add $^{+}$ or $^{-}$ for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 8

1. $\displaystyle \lim_{x\to0} t(x)=0$
2. $\displaystyle t(0)=3$
3. $\displaystyle \lim_{x\to0^+} t(x)=3$ (correct answer)
4. $\displaystyle \lim_{x\to0^-} t(x)=3$
5. $\displaystyle \lim_{x\to0^+} t(x)=0$

Explanation: This piecewise function has t(x) = x² for x < 0 and t(x) = 3 for x > 0. The question asks specifically about right-hand behavior as x → 0. From the right (x > 0), we use t(x) = 3, so lim_{x→0^+} t(x) = 3. From the left (x < 0), we use t(x) = x², so lim_{x→0^-} t(x) = 0² = 0. The correct notation for the right-hand limit is lim_{x→0^+} t(x) = 3, properly indicating approach from the positive side. Option E incorrectly states the right-hand limit is 0, which would be the left-hand limit. A common error is confusing which formula applies for each direction. Limit notation checklist: x → 0^+ means x > 0 (approaching from the right), x → 0^- means x < 0 (approaching from the left), and match the correct piece of the function to each direction.

Question 9

1. $\displaystyle \lim_{x\to 2} n(x)=5$
2. $\displaystyle \lim_{x\to 2^+} n(x)=5$ (correct answer)
3. $\displaystyle \lim_{x\to 2^-} n(x)=5$
4. $\displaystyle n(2)=5$
5. $\displaystyle \lim_{x\to 5} n(x)=2$

Explanation: Graph n(x) to 5 from right at 2, left to 1. \lim_{x\to 2^+} n(x)=5 correct. Valid right. Error: \lim_{x\to 2} n(x)=5. n(2)=5 value. Specify. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 10

1. $\displaystyle \lim_{x\to 0^+} h(x)=-2$ (correct answer)
2. $\displaystyle \lim_{x\to 0^-} h(x)=-2$
3. $\displaystyle \lim_{x\to 0} h(x)=2$
4. $\displaystyle \lim_{x\to -2} h(x)=0$
5. $\displaystyle \lim_{x\to 0} h(x)=h(0)$

Explanation: One-sided limit notation specifies direction using + for right or - for left, which is crucial when behavior differs on each side. The table shows h(x) approaching -2 only from the right as x nears 0, so \lim_{x \to 0^+} h(x) = -2 accurately represents this. This notation is valid as it matches the directional approach described. A common symbolic error is omitting the direction, like \lim_{x \to 0} h(x) = -2, assuming two-sided without evidence. Another mistake is confusing the limit with function equality, such as \lim_{x \to 0} h(x) = h(0). Always check if the data specifies one side or both. Transferable notation checklist: 1. Write as \lim_{x \to a} f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

Question 11

1. $\lim_{x\to5} u(x)=-3$
2. $\lim_{x\to5} u(x)=6$ (correct answer)
3. $\lim_{x\to5^-} u(x)=5$
4. $\lim_{x\to5^+} u(x)=-6$
5. $\lim_{x\to5} u(5)=-3$

Explanation: The function $u(x) = \frac{(x-5)(x+1)}{x-5}$ simplifies to $u(x) = x+1$ for all $x \neq 5$ by canceling the common factor $(x-5)$. As $x$ approaches 5, the simplified function approaches $5+1=6$, regardless of the assigned value $u(5)=-3$. The correct limit notation is $\lim_{x\to 5} u(x) = 6$. This illustrates a key principle: removable discontinuities don't affect limits—the limit exists even though the original expression is undefined at $x=5$. A common error is thinking the limit must equal the assigned function value, or writing $\lim_{x\to 5} u(5)$, which incorrectly evaluates the function inside the limit. Notation checklist: (1) $\lim$ symbol, (2) $x\to 5$, (3) $u(x)$ without substitution, (4) equals 6.

Question 12

1. $\lim_{x\to0} s(x)=0$
2. $\lim_{x\to0^+} s(x)=1$ (correct answer)
3. $\lim_{x\to0^-} s(x)=1$
4. $\lim_{x\to0^+} s(x)=-1$
5. $\lim_{x\to0} s(0)=0$

Explanation: The function $s(x) = \frac{|x|}{x}$ equals $\frac{x}{x} = 1$ when $x>0$ and $\frac{-x}{x} = -1$ when $x<0$. For the right-hand limit as $x\to 0^+$, we consider positive values of $x$ approaching 0, where $s(x) = 1$. Therefore, $\lim_{x\to 0^+} s(x) = 1$ is the correct notation. The assigned value $s(0)=0$ is irrelevant to the limit calculation—limits describe behavior near a point, not at it. A common error is writing $\lim_{x\to 0^+} s(0)$, which incorrectly substitutes 0 into the function within the limit notation. Note that the left-hand limit would be -1, so the two-sided limit doesn't exist. Notation checklist: (1) $\lim$ symbol, (2) $x\to 0^+$ for right approach, (3) $s(x)$ not $s(0)$, (4) equals 1.

Question 13

1. $\displaystyle q(3)=6$
2. $\displaystyle \lim_{x\to3} q(x)=6$ (correct answer)
3. $\displaystyle \lim_{x\to6} q(x)=3$
4. $\displaystyle \lim_{x\to3^-} q(x)=5.98$
5. $\displaystyle \lim_{x\to3^+} q(x)=6.02$

Explanation: The table shows q(x) values approaching 6 as x approaches 3 from both sides: from the left (2.9 → 5.98, 2.99 → 5.998) and from the right (3.01 → 6.002, 3.1 → 6.02). Since the values approach 6 from both directions, the two-sided limit exists and equals 6. The correct notation is lim_{x→3} q(x) = 6, using the standard two-sided limit notation without directional superscripts. Option A incorrectly states q(3) = 6, which is a function value, not a limit statement. A common error is using overly specific decimal values (like 5.98 or 6.02) instead of recognizing the pattern approaching 6. Limit notation checklist: use lim_{x→a} for two-sided limits, omit superscripts when approaching from both sides, and identify the limiting value from the pattern in the table.

Question 14

1. $\displaystyle \lim_{x\to4} r(x)=\frac14$ (correct answer)
2. $\displaystyle r(4)=\frac14$
3. $\displaystyle \lim_{x\to2} r(x)=\frac14$
4. $\displaystyle \lim_{x\to4^-} r(x)=-\frac14$
5. $\displaystyle \lim_{x=4} r(x)=\frac14$

Explanation: The function r(x) = (√x - 2)/(x - 4) is undefined at x = 4, but we can find the limit by rationalizing. Multiplying by (√x + 2)/(√x + 2), we get r(x) = (x - 4)/[(x - 4)(√x + 2)] = 1/(√x + 2) for x ≠ 4. As x approaches 4, r(x) approaches 1/(√4 + 2) = 1/(2 + 2) = 1/4. The correct notation is lim_{x→4} r(x) = 1/4, using the two-sided limit since the simplified form approaches the same value from both directions. Option E incorrectly uses x = 4 instead of x → 4, which is improper limit notation. A common error is forgetting to simplify before evaluating the limit. Limit notation checklist: use → not = in limits, simplify indeterminate forms before evaluating, and use two-sided notation when both one-sided limits agree.

Question 15

1. $\displaystyle \lim_{x\to1^-} p(x)=-\infty$ (correct answer)
2. $\displaystyle \lim_{x\to1} p(x)=0$
3. $\displaystyle \lim_{x\to1^+} p(x)=-\infty$
4. $\displaystyle p(1)=-\infty$
5. $\displaystyle \lim_{x\to1^-} p(x)=+\infty$

Explanation: For this piecewise function, when x < 1, p(x) = 1/(x - 1), and as x approaches 1 from the left, the denominator (x - 1) approaches 0 through negative values. Since we're dividing 1 by increasingly small negative numbers, p(x) approaches -∞. The correct notation is lim_{x→1^-} p(x) = -∞, which properly indicates both the direction of approach (from the left) and the infinite behavior. Note that p(1) is undefined, so option D is incorrect notation. A common error is confusing the signs: as x → 1^-, we have x - 1 < 0, making 1/(x - 1) negative. Limit notation checklist: use lim notation for limits (not function notation), include direction superscripts for one-sided limits, and carefully track signs when dealing with infinite limits.

Question 16

1. $\displaystyle \lim_{x\to -2} q(x)=0$ (correct answer)
2. $\displaystyle q(-2)=0$
3. $\displaystyle \lim_{x\to 0} q(x)=-2$
4. $\displaystyle \lim_{x\to -2^+} q(x)=2$
5. $\displaystyle \lim_{x\to -2} q(-2)=0$

Explanation: Limit notation describes functional behavior near a point, with the table indicating q(x) approaches 0 as x nears -2. Thus, \lim_{x\to -2} q(x)=0 represents this two-sided approach. This is valid as it ignores q(-2) and focuses on vicinity. A common error is \lim_{x\to -2} q(-2)=0, blending notation improperly. q(-2)=0 denotes value at -2, not limit. Limits exist independently of point values. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 17

1. $\displaystyle c(4)=-1$
2. $\displaystyle \lim_{x\to 4} c(x)=-1$ (correct answer)
3. $\displaystyle \lim_{x\to -1} c(x)=4$
4. $\displaystyle \lim_{x\to 4^-} c(x)=1$
5. $\displaystyle \lim_{x\to 4} c(4)=-1$

Explanation: Table suggests c(x) to -1 at x=4. \lim_{x\to 4} c(x)=-1 represents. Valid both sides. Error: \lim_{x\to 4} c(4)=-1. c(4)=-1 value. Separate. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Question 18

1. $\displaystyle \lim_{x\to 3} g(x)=10$
2. $\displaystyle \lim_{x\to 10} g(x)=3$
3. $\displaystyle \lim_{x\to 3} g(x)=6$ (correct answer)
4. $\displaystyle \lim_{x\to 3^-} g(x)=10$
5. $\displaystyle \lim_{g(x)\to 3} x=6$

Explanation: Limit notation for rational functions often involves simplifying expressions to find the approaching value, ignoring the function's defined value at the point. For g(x) = ($x^2$ - 9)/(x - 3) simplified to x + 3 for x ≠ 3, the limit as x approaches 3 is 6, despite g(3) = 10, so \lim_{x \to 3} g(x) = 6 is correct. This is valid because limits consider behavior near the point, not at it. A common error is using the redefined value, like \lim_{x \to 3} g(x) = 10. Another mistake is reversing the limit, such as \lim_{x \to 10} g(x) = 3 or \lim_{g(x) \to 3} x = 6. The two-sided limit applies here as the function approaches the same value from both sides. Transferable notation checklist: 1. Write as \lim_{x \to a} f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

Question 19

1. $\lim_{x\to4^+} r(x)=2$ (correct answer)
2. $\lim_{x\to4^-} r(x)=3$
3. $\lim_{x\to4} r(x)=4$
4. $r(4)=3$
5. $\lim_{x\to4^+} r(4)=2$

Explanation: The function $r$ is defined as $r(x)=2$ for $x\leq 4$ and $r(x)=2+(x-4)$ for $x>4$. For the right-hand limit as $x\to 4^+$, we use the formula for $x>4$: as $x$ approaches 4 from the right, $(x-4)$ approaches 0, so $r(x)$ approaches $2+0=2$. The correct notation is $\lim_{x\to 4^+} r(x) = 2$, where the superscript plus indicates we only consider values greater than 4. Notice that both one-sided limits equal 2, making the function continuous at $x=4$. A common error is writing $\lim_{x\to 4^+} r(4)$, which incorrectly evaluates the function at 4 rather than describing the limiting behavior. Notation checklist: (1) $\lim$ symbol, (2) $x\to 4^+$ for right approach, (3) $r(x)$ not $r(4)$, (4) equals 2.

Question 20

1. $\lim_{x\to-1} q(x)=10$
2. $\lim_{x\to-1} q(x)=-2$ (correct answer)
3. $\lim_{x\to-1^+} q(x)=2$
4. $\lim_{x\to-1} q(-1)=10$
5. $\lim_{x\to-1^-} q(x)=10$

Explanation: The function $q(x) = \frac{x^2-1}{x+1}$ can be simplified by factoring the numerator as $(x-1)(x+1)$, giving us $q(x) = \frac{(x-1)(x+1)}{x+1} = x-1$ for all $x \neq -1$. As $x$ approaches $-1$, the simplified function approaches $-1-1 = -2$, regardless of the assigned value $q(-1)=10$. The correct limit notation is $\lim_{x\to -1} q(x) = -2$. A common error is thinking the limit must equal the function value at the point, but limits describe nearby behavior, not the value at the point itself. Another mistake is writing $\lim_{x\to -1} q(-1)$, which incorrectly evaluates the function inside the limit. Notation checklist: (1) $\lim$ symbol, (2) $x\to -1$, (3) $q(x)$ without substitution, (4) equals -2.