AP Calculus AB Flashcards: Algebraic Properties Of Limits

Study Algebraic Properties Of Limits in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Algebraic Properties Of Limits

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QUESTION

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Find $\text{lim}_{x \to -2} \frac{x^2 + 4x + 4}{x + 2}$ . Use algebraic properties.

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ANSWER

The limit is $-4$ . Factor perfect square: $\frac{(x+2)^2}{x+2} = x+2$ , then substitute.

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What this deck covers

This deck focuses on Algebraic Properties Of Limits, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Find $\text{lim}_{x \to -2} \frac{x^2 + 4x + 4}{x + 2}$ . Use algebraic properties.

Answer: The limit is $-4$ . Factor perfect square: $\frac{(x+2)^2}{x+2} = x+2$ , then substitute.

Flashcard 2: Find $\text{lim}_{x \to 1} \frac{x^2 - 1}{x^2 + x - 2}$ . Use algebraic properties.

Answer: The limit is $0$ . Factor both: $\frac{(x-1)(x+1)}{(x-1)(x+2)} = \frac{x+1}{x+2}$ , then substitute.

Flashcard 3: What is the limit property for a constant multiple: $\text{If }\text{lim}_{x \to a} f(x) = L$ , what is $\text{lim}_{x \to a} [cf(x)]$ ?

Answer: The limit is $cL$ . Constant multiple property: factor out constants.

Flashcard 4: Find $\text{lim}_{x \to 4} \frac{x^2 - 16}{x - 4}$ . Use algebraic properties.

Answer: The limit is $8$ . Factor difference of squares: $\frac{(x+4)(x-4)}{x-4} = x+4$ .

Flashcard 5: What is the limit of a power function: $\text{If }\lim_{x \to a} f(x) = L$ , what is $\lim_{x \to a} [f(x)]^n$ ?

Answer: The limit is $L^n$ . Power property: raise the limit to the same power.

Flashcard 6: Find $\text{lim}_{x \to -1} (x^3 + 4x^2 + x - 6)$ . Use algebraic properties.

Answer: The limit is $-2$ . Direct substitution: $(-1)^3 + 4(-1)^2 + (-1) - 6 = -2$ .

Flashcard 7: What is the limit of a power function: $\text{If }\text{lim}_{x \to a} f(x) = L$ , what is $\text{lim}_{x \to a} [f(x)]^n$ ?

Answer: The limit is $L^n$ . Power property: raise the limit to the same power.

Flashcard 8: Find $\text{lim}_{x \to 1} (3x^3 - 2x^2 + x - 5)$ . Use algebraic properties.

Answer: The limit is $-3$ . Direct substitution: $3(1)^3 - 2(1)^2 + 1 - 5 = -3$ .

Flashcard 9: Find $\text{lim}_{x \to 1} (3x^3 - 2x^2 + x - 5)$ . Use algebraic properties.

Answer: The limit is $-3$ . Direct substitution: $3(1)^3 - 2(1)^2 + 1 - 5 = -3$ .

Flashcard 10: Find $\text{lim}_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}$ . Use algebraic properties.

Answer: The limit is $-1$ . Factor: $\frac{(x-2)(x-3)}{x-2} = x-3$ , then substitute.

Flashcard 11: Find $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{\text{sin}(x)}$ . Use algebraic properties.

Answer: The limit is $3$ . Rewrite as $\frac{\sin(3x)}{3x} \cdot \frac{3x}{x} \cdot \frac{x}{\sin(x)} = 1 \cdot 3 \cdot 1$ .

Flashcard 12: Find $\text{lim}_{x \to 0} x^2 \text{cos}(x)$ . Use algebraic properties.

Answer: The limit is $0$ . Product of limits: $\text{lim } x^2 \times \text{lim } \text{cos}(x) = 0 \times 1$ .

Flashcard 13: What is the limit of a constant function $f(x) = c$ as $x$ approaches any value $a$ ?

Answer: The limit is $c$ . Constants remain unchanged regardless of the input value.

Flashcard 14: Find $\text{lim}_{x \to 4} \frac{x^2 - 16}{x - 4}$ . Use algebraic properties.

Answer: The limit is $8$ . Factor difference of squares: $\frac{(x+4)(x-4)}{x-4} = x+4$ .

Flashcard 15: Find $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$ . Use algebraic properties.

Answer: The limit is $4$ . Factor and cancel: $\frac{(x+2)(x-2)}{x-2} = x+2$ , then substitute.

Flashcard 16: Find $\text{lim}_{x \to -1} (x^3 + 4x^2 + x - 6)$ . Use algebraic properties.

Answer: The limit is $-2$ . Direct substitution: $(-1)^3 + 4(-1)^2 + (-1) - 6 = -2$ .

Flashcard 17: Find $\text{lim}_{x \to 4} \frac{x - 4}{\text{sqrt}(x) - 2}$ . Use algebraic properties.

Answer: The limit is $4$ . Rationalize by multiplying by $\frac{\sqrt{x}+2}{\sqrt{x}+2}$ .

Flashcard 18: What is the limit of $f(x) = x$ as $x$ approaches $a$ ?

Answer: The limit is $a$ . The identity function equals its input value.

Flashcard 19: What is the limit property for a constant multiple: $\text{If }\text{lim}_{x \to a} f(x) = L$ , what is $\text{lim}_{x \to a} [cf(x)]$ ?

Answer: The limit is $cL$ . Constant multiple property: factor out constants.

Flashcard 20: Find $\text{lim}_{x \to 0} \frac{x}{\text{sin}(x)}$ . Use known results.

Answer: The limit is $1$ . Reciprocal of the standard limit $\sin(x)/x = 1$ .

Flashcard 21: If $\text{lim}_{x \to a} f(x) = L$ , what is $\text{lim}_{x \to a} (f(x) + c)$ ?

Answer: The limit is $L + c$ . Sum property: limit of sum equals sum of limits.

Flashcard 22: What is the limit of $f(x) = x$ as $x$ approaches $a$ ?

Answer: The limit is $a$ . The identity function equals its input value.

Flashcard 23: Find $\text{lim}_{x \to 4} \frac{x - 4}{\text{sqrt}(x) - 2}$ . Use algebraic properties.

Answer: The limit is $4$ . Rationalize by multiplying by $\frac{\sqrt{x}+2}{\sqrt{x}+2}$ .

Flashcard 24: Find $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$ . Use algebraic properties.

Answer: The limit is $3$ . Factor difference of cubes: $\frac{(x-1)(x^2+x+1)}{x-1} = x^2+x+1$ .

Flashcard 25: Find $\text{lim}_{x \to 0} \frac{x}{\text{sin}(x)}$ . Use known results.

Answer: The limit is $1$ . Reciprocal of the standard limit $\sin(x)/x = 1$ .

Flashcard 26: Find $\text{lim}_{x \to 1} \frac{x^2 - 1}{x^2 + x - 2}$ . Use algebraic properties.

Answer: The limit is $0$ . Factor both: $\frac{(x-1)(x+1)}{(x-1)(x+2)} = \frac{x+1}{x+2}$ , then substitute.

Flashcard 27: Find $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$ . Use algebraic properties.

Answer: The limit is $3$ . Factor difference of cubes: $\frac{(x-1)(x^2+x+1)}{x-1} = x^2+x+1$ .

Flashcard 28: Find $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$ . Use known results.

Answer: The limit is $2$ . Use $\sin(2x) = 2\sin(x)\cos(x)$ and known limit $\sin(x)/x = 1$ .

Flashcard 29: Find $\text{lim}_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}$ . Use algebraic properties.

Answer: The limit is $-1$ . Factor: $\frac{(x-2)(x-3)}{x-2} = x-3$ , then substitute.

Flashcard 30: Find $\text{lim}_{x \to 2} \frac{x^3 - 8}{x - 2}$ . Use algebraic properties.

Answer: The limit is $12$ . Factor difference of cubes: $\frac{(x-2)(x^2+2x+4)}{x-2} = x^2+2x+4$ .

Flashcard 31: Find $\text{lim}_{x \to 0} x^2 \text{cos}(x)$ . Use algebraic properties.

Answer: The limit is $0$ . Product of limits: $\text{lim } x^2 \times \text{lim } \text{cos}(x) = 0 \times 1$ .

Flashcard 32: What is the algebraic property of limits for the absolute value function?

Answer: $\lim_{x \to a} |f(x)| = |\lim_{x \to a} f(x)|$ . Absolute value of the limit equals limit of absolute value.

Flashcard 33: Find $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{\text{sin}(x)}$ . Use algebraic properties.

Answer: The limit is $3$ . Rewrite as $\frac{\sin(3x)}{3x} \cdot \frac{3x}{x} \cdot \frac{x}{\sin(x)} = 1 \cdot 3 \cdot 1$ .

Flashcard 34: If $\text{lim}_{x \to a} f(x) = L$ , what is $\text{lim}_{x \to a} (f(x) + c)$ ?

Answer: The limit is $L + c$ . Sum property: limit of sum equals sum of limits.

Flashcard 35: Find $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$ . Use algebraic properties.

Answer: The limit is $2$ . Factor difference of squares: $\frac{(x+1)(x-1)}{x-1} = x+1$ .

Flashcard 36: Find $\text{lim}_{x \to 2} \frac{\text{sqrt}(x+2) - 2}{x - 2}$ . Use algebraic properties.

Answer: The limit is $\frac{1}{4}$ . Rationalize by multiplying by $\frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$ .

Flashcard 37: Find $\text{lim}_{x \to 2} \frac{x^3 - 8}{x - 2}$ . Use algebraic properties.

Answer: The limit is $12$ . Factor difference of cubes: $\frac{(x-2)(x^2+2x+4)}{x-2} = x^2+2x+4$ .

Flashcard 38: Find $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$ . Use algebraic properties.

Answer: The limit is $4$ . Factor and cancel: $\frac{(x+2)(x-2)}{x-2} = x+2$ , then substitute.

Flashcard 39: Find $\text{lim}_{x \to -2} \frac{x^2 + 4x + 4}{x + 2}$ . Use algebraic properties.

Answer: The limit is $-4$ . Factor perfect square: $\frac{(x+2)^2}{x+2} = x+2$ , then substitute.

Flashcard 40: What is the limit of a constant function $f(x) = c$ as $x$ approaches any value $a$ ?

Answer: The limit is $c$ . Constants remain unchanged regardless of the input value.

Flashcard 41: Find $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$ . Use algebraic properties.

Answer: The limit is $2$ . Factor difference of squares: $\frac{(x+1)(x-1)}{x-1} = x+1$ .

Flashcard 42: What is the algebraic property of limits for the absolute value function?

Answer: $\lim_{x \to a} |f(x)| = |\lim_{x \to a} f(x)|$ . Absolute value of the limit equals limit of absolute value.

Flashcard 43: Find $\text{lim}_{x \to 2} \frac{\text{sqrt}(x+2) - 2}{x - 2}$ . Use algebraic properties.

Answer: The limit is $\frac{1}{4}$ . Rationalize by multiplying by $\frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$ .

Flashcard 44: Find $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$ . Use algebraic properties.

Answer: The limit is $6$ . Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$ .

Flashcard 45: Find $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$ . Use known results.

Answer: The limit is $2$ . Use $\sin(2x) = 2\sin(x)\cos(x)$ and known limit $\sin(x)/x = 1$ .

Flashcard 46: Find $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$ . Use algebraic properties.

Answer: The limit is $6$ . Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$ .

AP Calculus AB Flashcards: Algebraic Properties Of Limits

Algebraic Properties Of Limits

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Flashcard 1: Find limx→−2x2+4x+4x+2\text{lim}_{x \to -2} \frac{x^2 + 4x + 4}{x + 2}limx→−2​x+2x2+4x+4​. Use algebraic properties.

Flashcard 2: Find limx→1x2−1x2+x−2\text{lim}_{x \to 1} \frac{x^2 - 1}{x^2 + x - 2}limx→1​x2+x−2x2−1​. Use algebraic properties.

Flashcard 3: What is the limit property for a constant multiple: If limx→af(x)=L\text{If }\text{lim}_{x \to a} f(x) = LIf limx→a​f(x)=L, what is limx→a[cf(x)]\text{lim}_{x \to a} [cf(x)]limx→a​[cf(x)]?

Flashcard 4: Find limx→4x2−16x−4\text{lim}_{x \to 4} \frac{x^2 - 16}{x - 4}limx→4​x−4x2−16​. Use algebraic properties.

Flashcard 5: What is the limit of a power function: If lim⁡x→af(x)=L\text{If }\lim_{x \to a} f(x) = LIf limx→a​f(x)=L, what is lim⁡x→a[f(x)]n\lim_{x \to a} [f(x)]^nlimx→a​[f(x)]n?

Flashcard 6: Find limx→−1(x3+4x2+x−6)\text{lim}_{x \to -1} (x^3 + 4x^2 + x - 6)limx→−1​(x3+4x2+x−6). Use algebraic properties.

Flashcard 7: What is the limit of a power function: If limx→af(x)=L\text{If }\text{lim}_{x \to a} f(x) = LIf limx→a​f(x)=L, what is limx→a[f(x)]n\text{lim}_{x \to a} [f(x)]^nlimx→a​[f(x)]n?

Flashcard 8: Find limx→1(3x3−2x2+x−5)\text{lim}_{x \to 1} (3x^3 - 2x^2 + x - 5)limx→1​(3x3−2x2+x−5). Use algebraic properties.

Flashcard 9: Find limx→1(3x3−2x2+x−5)\text{lim}_{x \to 1} (3x^3 - 2x^2 + x - 5)limx→1​(3x3−2x2+x−5). Use algebraic properties.

Flashcard 10: Find limx→2x2−5x+6x−2\text{lim}_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}limx→2​x−2x2−5x+6​. Use algebraic properties.

Flashcard 11: Find limx→0sin(3x)sin(x)\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{\text{sin}(x)}limx→0​sin(x)sin(3x)​. Use algebraic properties.

Flashcard 12: Find limx→0x2cos(x)\text{lim}_{x \to 0} x^2 \text{cos}(x)limx→0​x2cos(x). Use algebraic properties.

Flashcard 13: What is the limit of a constant function f(x)=cf(x) = cf(x)=c as xxx approaches any value aaa?

Flashcard 14: Find limx→4x2−16x−4\text{lim}_{x \to 4} \frac{x^2 - 16}{x - 4}limx→4​x−4x2−16​. Use algebraic properties.

Flashcard 15: Find limx→2x2−4x−2\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}limx→2​x−2x2−4​. Use algebraic properties.

Flashcard 16: Find limx→−1(x3+4x2+x−6)\text{lim}_{x \to -1} (x^3 + 4x^2 + x - 6)limx→−1​(x3+4x2+x−6). Use algebraic properties.

Flashcard 17: Find limx→4x−4sqrt(x)−2\text{lim}_{x \to 4} \frac{x - 4}{\text{sqrt}(x) - 2}limx→4​sqrt(x)−2x−4​. Use algebraic properties.

Flashcard 18: What is the limit of f(x)=xf(x) = xf(x)=x as xxx approaches aaa?

Flashcard 19: What is the limit property for a constant multiple: If limx→af(x)=L\text{If }\text{lim}_{x \to a} f(x) = LIf limx→a​f(x)=L, what is limx→a[cf(x)]\text{lim}_{x \to a} [cf(x)]limx→a​[cf(x)]?

Flashcard 20: Find limx→0xsin(x)\text{lim}_{x \to 0} \frac{x}{\text{sin}(x)}limx→0​sin(x)x​. Use known results.

Flashcard 21: If limx→af(x)=L\text{lim}_{x \to a} f(x) = Llimx→a​f(x)=L, what is limx→a(f(x)+c)\text{lim}_{x \to a} (f(x) + c)limx→a​(f(x)+c)?

Flashcard 22: What is the limit of f(x)=xf(x) = xf(x)=x as xxx approaches aaa?

Flashcard 23: Find limx→4x−4sqrt(x)−2\text{lim}_{x \to 4} \frac{x - 4}{\text{sqrt}(x) - 2}limx→4​sqrt(x)−2x−4​. Use algebraic properties.

Flashcard 24: Find limx→1x3−1x−1\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}limx→1​x−1x3−1​. Use algebraic properties.

Flashcard 25: Find limx→0xsin(x)\text{lim}_{x \to 0} \frac{x}{\text{sin}(x)}limx→0​sin(x)x​. Use known results.

Flashcard 26: Find limx→1x2−1x2+x−2\text{lim}_{x \to 1} \frac{x^2 - 1}{x^2 + x - 2}limx→1​x2+x−2x2−1​. Use algebraic properties.

Flashcard 27: Find limx→1x3−1x−1\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}limx→1​x−1x3−1​. Use algebraic properties.

Flashcard 28: Find limx→0sin(2x)x\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}limx→0​xsin(2x)​. Use known results.

Flashcard 29: Find limx→2x2−5x+6x−2\text{lim}_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}limx→2​x−2x2−5x+6​. Use algebraic properties.

Flashcard 30: Find limx→2x3−8x−2\text{lim}_{x \to 2} \frac{x^3 - 8}{x - 2}limx→2​x−2x3−8​. Use algebraic properties.

Flashcard 31: Find limx→0x2cos(x)\text{lim}_{x \to 0} x^2 \text{cos}(x)limx→0​x2cos(x). Use algebraic properties.