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AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

Study Defining Limits And Using Limit Notation in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

What this deck covers

This deck focuses on Defining Limits And Using Limit Notation, 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.

AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

/ 30

0 reviewed

0% Complete

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QUESTION

How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Tap or drag to reveal answer

ANSWER

$\lim_{{x \to a^-}} f(x)$ . The $-$ superscript indicates approaching from values less than $a$ .

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Answer: $\lim_{{x \to a^-}} f(x)$ . The $-$ superscript indicates approaching from values less than $a$ .

Flashcard 2: Identify the limit law for the constant multiple of a function.

Answer: $\lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x)$ . Constants can be factored out of limit expressions.

Flashcard 3: What is the limit of $f(x) = c$ as $x$ approaches any $a$ ?

Answer: $c$ . Constant functions have limits equal to the constant value everywhere.

Flashcard 4: State the limit law for the product of two functions.

Answer: $\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$ . Limit of product equals product of limits when both individual limits exist.

Flashcard 5: What does it mean if $\lim_{{x \to a}} f(x) = L$ ?

Answer: As $x$ approaches $a$ , $f(x)$ approaches $L$ . The function value gets arbitrarily close to $L$ as input approaches $a$ .

Flashcard 6: Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Answer:

This is a fundamental trigonometric limit used in calculus.

Flashcard 7: What is the condition for the existence of a two-sided limit?

Answer: Both one-sided limits must exist and be equal. When $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x)$ , the two-sided limit exists.

Flashcard 8: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.

Answer: $\infty$ . As $x$ approaches $0^+$ , $\frac{1}{x}$ becomes arbitrarily large and positive.

Flashcard 9: How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Answer: $\lim_{{x \to a^+}} f(x)$ . The $+$ superscript indicates approaching from values greater than $a$ .

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$ .

Answer:

Divide numerator and denominator by $x^3$ ; limit of $\frac{3x^3}{x^3} = 3$ .

Flashcard 11: State the limit law for powers of a function.

Answer: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$ . Limit of power equals power of limit when the base limit exists.

Flashcard 12: State the limit law for the difference of two functions.

Answer: $\lim_{{x \to a}} [f(x) - g(x)] = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)$ . Limit of difference equals difference of limits when both individual limits exist.

Flashcard 13: Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$ .

Answer:

Direct substitution: $(-1)^3 + 2(-1)^2 + (-1) = -1 + 2 - 1 = 0$ .

Flashcard 14: What is the definition of a limit in calculus?

Answer: The value a function approaches as the input approaches a point. This captures the fundamental concept of approaching a value without necessarily reaching it.

Flashcard 15: What is a one-sided limit?

Answer: The limit of a function as $x$ approaches a point from one side, either the left or the right. Considers approach from only left ( $a^-$ ) or right ( $a^+$ ) side of the point.

Flashcard 16: What is the limit notation for $f(x)$ as $x$ approaches infinity?

Answer: $\lim_{{x \to \infty}} f(x)$ . Examines function behavior as input values become arbitrarily large.

Flashcard 17: How is an infinite limit denoted?

Answer: $\lim_{x \to a} f(x) = \infty$ or $-\infty$ . Uses infinity symbol to show unbounded growth in positive or negative direction.

Flashcard 18: State the limit law for the quotient of two functions.

Answer: $\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)}$ , if $\lim_{{x \to a}} g(x) \neq 0$ . Limit of quotient equals quotient of limits when denominator limit is nonzero.

Flashcard 19: Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$ .

Answer: $\frac{5}{2}$ . Divide by highest power $x^2$ : $\frac{5-\frac{4}{x}}{2+\frac{3}{x^2}} \to \frac{5}{2}$ .

Flashcard 20: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.

Answer: $-\infty$ . As $x$ approaches $0^-$ , $\frac{1}{x}$ becomes arbitrarily large and negative.

Flashcard 21: If $\lim_{{x \to a}} f(x) = L$ , what is the relationship between $f(x)$ and $L$ ?

Answer: As $x$ approaches $a$ , $f(x)$ gets arbitrarily close to $L$ . The function output approaches $L$ but doesn't necessarily equal $L$ at $x = a$ .

Flashcard 22: Evaluate $\lim_{{x \to 3}} (x^2 - 9)$ .

Answer:

Direct substitution: $(3)^2 - 9 = 9 - 9 = 0$ .

Flashcard 23: What does continuity at a point $x = a$ require?

Answer: $f(a)$ is defined, $\lim_{{x \to a}} f(x)$ exists, and $\lim_{{x \to a}} f(x) = f(a)$ . Three conditions ensure no gaps, jumps, or undefined points.

Flashcard 24: What is the limit of $f(x) = x$ as $x$ approaches $a$ ?

Answer: $a$ . The identity function $f(x) = x$ approaches the point value.

Flashcard 25: How is the limit of a function as $x$ approaches $a$ denoted?

Answer: $\lim_{{x \to a}} f(x)$ . Standard mathematical notation using $\lim$ with subscript showing the approach.

Flashcard 26: What is an infinite limit?

Answer: A limit where $f(x)$ increases or decreases without bound as $x$ approaches a point. Function values grow without bound, approaching positive or negative infinity.

Flashcard 27: State the limit law for the sum of two functions.

Answer: $\lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)$ . Limit of sum equals sum of limits when both individual limits exist.

Flashcard 28: Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Answer:

Factor as $\frac{(x+1)(x-1)}{x-1}$ , cancel to get $x+1$ , then substitute $x=1$ .

Flashcard 29: Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$ .

Answer: $\infty$ . Polynomial with positive leading coefficient grows without bound as $x \to \infty$ .

Flashcard 30: What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$ ?

Answer: Right-hand limit and left-hand limit, respectively. These describe approaches from the positive and negative sides respectively.

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AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

Study Defining Limits And Using Limit Notation in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

What this deck covers

This deck focuses on Defining Limits And Using Limit Notation, 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.

AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

/ 30

0 reviewed

0% Complete

0 reviewing

QUESTION

How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Tap or drag to reveal answer

ANSWER

$\lim_{{x \to a^-}} f(x)$ . The $-$ superscript indicates approaching from values less than $a$ .

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Answer: $\lim_{{x \to a^-}} f(x)$ . The $-$ superscript indicates approaching from values less than $a$ .

Flashcard 2: Identify the limit law for the constant multiple of a function.

Answer: $\lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x)$ . Constants can be factored out of limit expressions.

Flashcard 3: What is the limit of $f(x) = c$ as $x$ approaches any $a$ ?

Answer: $c$ . Constant functions have limits equal to the constant value everywhere.

Flashcard 4: State the limit law for the product of two functions.

Answer: $\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$ . Limit of product equals product of limits when both individual limits exist.

Flashcard 5: What does it mean if $\lim_{{x \to a}} f(x) = L$ ?

Answer: As $x$ approaches $a$ , $f(x)$ approaches $L$ . The function value gets arbitrarily close to $L$ as input approaches $a$ .

Flashcard 6: Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Answer:

This is a fundamental trigonometric limit used in calculus.

Flashcard 7: What is the condition for the existence of a two-sided limit?

Answer: Both one-sided limits must exist and be equal. When $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x)$ , the two-sided limit exists.

Flashcard 8: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.

Answer: $\infty$ . As $x$ approaches $0^+$ , $\frac{1}{x}$ becomes arbitrarily large and positive.

Flashcard 9: How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Answer: $\lim_{{x \to a^+}} f(x)$ . The $+$ superscript indicates approaching from values greater than $a$ .

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$ .

Answer:

Divide numerator and denominator by $x^3$ ; limit of $\frac{3x^3}{x^3} = 3$ .

Flashcard 11: State the limit law for powers of a function.

Answer: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$ . Limit of power equals power of limit when the base limit exists.

Flashcard 12: State the limit law for the difference of two functions.

Answer: $\lim_{{x \to a}} [f(x) - g(x)] = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)$ . Limit of difference equals difference of limits when both individual limits exist.

Flashcard 13: Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$ .

Answer:

Direct substitution: $(-1)^3 + 2(-1)^2 + (-1) = -1 + 2 - 1 = 0$ .

Flashcard 14: What is the definition of a limit in calculus?

Answer: The value a function approaches as the input approaches a point. This captures the fundamental concept of approaching a value without necessarily reaching it.

Flashcard 15: What is a one-sided limit?

Answer: The limit of a function as $x$ approaches a point from one side, either the left or the right. Considers approach from only left ( $a^-$ ) or right ( $a^+$ ) side of the point.

Flashcard 16: What is the limit notation for $f(x)$ as $x$ approaches infinity?

Answer: $\lim_{{x \to \infty}} f(x)$ . Examines function behavior as input values become arbitrarily large.

Flashcard 17: How is an infinite limit denoted?

Answer: $\lim_{x \to a} f(x) = \infty$ or $-\infty$ . Uses infinity symbol to show unbounded growth in positive or negative direction.

Flashcard 18: State the limit law for the quotient of two functions.

Flashcard 19: Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$ .

Answer: $\frac{5}{2}$ . Divide by highest power $x^2$ : $\frac{5-\frac{4}{x}}{2+\frac{3}{x^2}} \to \frac{5}{2}$ .

Flashcard 20: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.

Answer: $-\infty$ . As $x$ approaches $0^-$ , $\frac{1}{x}$ becomes arbitrarily large and negative.

Flashcard 21: If $\lim_{{x \to a}} f(x) = L$ , what is the relationship between $f(x)$ and $L$ ?

Answer: As $x$ approaches $a$ , $f(x)$ gets arbitrarily close to $L$ . The function output approaches $L$ but doesn't necessarily equal $L$ at $x = a$ .

Flashcard 22: Evaluate $\lim_{{x \to 3}} (x^2 - 9)$ .

Answer:

Direct substitution: $(3)^2 - 9 = 9 - 9 = 0$ .

Flashcard 23: What does continuity at a point $x = a$ require?

Answer: $f(a)$ is defined, $\lim_{{x \to a}} f(x)$ exists, and $\lim_{{x \to a}} f(x) = f(a)$ . Three conditions ensure no gaps, jumps, or undefined points.

Flashcard 24: What is the limit of $f(x) = x$ as $x$ approaches $a$ ?

Answer: $a$ . The identity function $f(x) = x$ approaches the point value.

Flashcard 25: How is the limit of a function as $x$ approaches $a$ denoted?

Answer: $\lim_{{x \to a}} f(x)$ . Standard mathematical notation using $\lim$ with subscript showing the approach.

Flashcard 26: What is an infinite limit?

Answer: A limit where $f(x)$ increases or decreases without bound as $x$ approaches a point. Function values grow without bound, approaching positive or negative infinity.

Flashcard 27: State the limit law for the sum of two functions.

Answer: $\lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)$ . Limit of sum equals sum of limits when both individual limits exist.

Flashcard 28: Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Answer:

Factor as $\frac{(x+1)(x-1)}{x-1}$ , cancel to get $x+1$ , then substitute $x=1$ .

Flashcard 29: Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$ .

Answer: $\infty$ . Polynomial with positive leading coefficient grows without bound as $x \to \infty$ .

Flashcard 30: What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$ ?

Answer: Right-hand limit and left-hand limit, respectively. These describe approaches from the positive and negative sides respectively.

Opening subject page...

AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

What this deck covers

How to use these flashcards

AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

All flashcards

Flashcard 1: How is the left-hand limit of f(x)f(x)f(x) as xxx approaches aaa denoted?

Flashcard 2: Identify the limit law for the constant multiple of a function.

Flashcard 3: What is the limit of f(x)=cf(x) = cf(x)=c as xxx approaches any aaa?

Flashcard 4: State the limit law for the product of two functions.

Flashcard 5: What does it mean if lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L?

Flashcard 6: Evaluate lim⁡x→0sin⁡xx\lim_{{x \to 0}} \frac{\sin x}{x}limx→0​xsinx​.

Flashcard 7: What is the condition for the existence of a two-sided limit?

Flashcard 8: Identify the limit of 1x\frac{1}{x}x1​ as xxx approaches 000 from the right.

Flashcard 9: How is the right-hand limit of f(x)f(x)f(x) as xxx approaches aaa denoted?

Flashcard 10: Evaluate lim⁡x→∞3x3+x+2x3\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}limx→∞​x33x3+x+2​.

Flashcard 11: State the limit law for powers of a function.

Flashcard 12: State the limit law for the difference of two functions.

Flashcard 13: Evaluate lim⁡x→−1(x3+2x2+x)\lim_{{x \to -1}} (x^3 + 2x^2 + x)limx→−1​(x3+2x2+x).

Flashcard 14: What is the definition of a limit in calculus?

Flashcard 15: What is a one-sided limit?

Flashcard 16: What is the limit notation for f(x)f(x)f(x) as xxx approaches infinity?

Flashcard 17: How is an infinite limit denoted?

Flashcard 18: State the limit law for the quotient of two functions.

Flashcard 19: Evaluate lim⁡x→∞5x2−4x2x2+3\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}limx→∞​2x2+35x2−4x​.

Flashcard 20: Identify the limit of 1x\frac{1}{x}x1​ as xxx approaches 000 from the left.

Flashcard 21: If lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L, what is the relationship between f(x)f(x)f(x) and LLL?

Flashcard 22: Evaluate lim⁡x→3(x2−9)\lim_{{x \to 3}} (x^2 - 9)limx→3​(x2−9).

Flashcard 23: What does continuity at a point x=ax = ax=a require?

Flashcard 24: What is the limit of f(x)=xf(x) = xf(x)=x as xxx approaches aaa?

Flashcard 25: How is the limit of a function as xxx approaches aaa denoted?

Flashcard 26: What is an infinite limit?

Flashcard 27: State the limit law for the sum of two functions.

Flashcard 28: Evaluate lim⁡x→1x2−1x−1\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}limx→1​x−1x2−1​.

Flashcard 29: Evaluate lim⁡x→∞(2x2+3x+1)\lim_{{x \to \infty}} (2x^2 + 3x + 1)limx→∞​(2x2+3x+1).

Flashcard 30: What is the formal name for lim⁡x→a+f(x)\lim_{{x \to a^+}} f(x)limx→a+​f(x) and lim⁡x→a−f(x)\lim_{{x \to a^-}} f(x)limx→a−​f(x)?

Opening subject page...

AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

What this deck covers

How to use these flashcards

AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

All flashcards

Flashcard 1: How is the left-hand limit of f(x)f(x)f(x) as xxx approaches aaa denoted?

Flashcard 2: Identify the limit law for the constant multiple of a function.

Flashcard 3: What is the limit of f(x)=cf(x) = cf(x)=c as xxx approaches any aaa?

Flashcard 4: State the limit law for the product of two functions.

Flashcard 5: What does it mean if lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L?

Flashcard 6: Evaluate lim⁡x→0sin⁡xx\lim_{{x \to 0}} \frac{\sin x}{x}limx→0​xsinx​.

Flashcard 7: What is the condition for the existence of a two-sided limit?

Flashcard 8: Identify the limit of 1x\frac{1}{x}x1​ as xxx approaches 000 from the right.

Flashcard 9: How is the right-hand limit of f(x)f(x)f(x) as xxx approaches aaa denoted?

Flashcard 10: Evaluate lim⁡x→∞3x3+x+2x3\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}limx→∞​x33x3+x+2​.

Flashcard 11: State the limit law for powers of a function.

Flashcard 12: State the limit law for the difference of two functions.

Flashcard 13: Evaluate lim⁡x→−1(x3+2x2+x)\lim_{{x \to -1}} (x^3 + 2x^2 + x)limx→−1​(x3+2x2+x).

Flashcard 14: What is the definition of a limit in calculus?

Flashcard 15: What is a one-sided limit?

Flashcard 16: What is the limit notation for f(x)f(x)f(x) as xxx approaches infinity?

Flashcard 17: How is an infinite limit denoted?

Flashcard 18: State the limit law for the quotient of two functions.

Flashcard 19: Evaluate lim⁡x→∞5x2−4x2x2+3\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}limx→∞​2x2+35x2−4x​.

Flashcard 20: Identify the limit of 1x\frac{1}{x}x1​ as xxx approaches 000 from the left.

Flashcard 21: If lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L, what is the relationship between f(x)f(x)f(x) and LLL?

Flashcard 22: Evaluate lim⁡x→3(x2−9)\lim_{{x \to 3}} (x^2 - 9)limx→3​(x2−9).

Flashcard 23: What does continuity at a point x=ax = ax=a require?

Flashcard 24: What is the limit of f(x)=xf(x) = xf(x)=x as xxx approaches aaa?

Flashcard 25: How is the limit of a function as xxx approaches aaa denoted?

Flashcard 26: What is an infinite limit?

Flashcard 27: State the limit law for the sum of two functions.

Flashcard 28: Evaluate lim⁡x→1x2−1x−1\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}limx→1​x−1x2−1​.

Flashcard 29: Evaluate lim⁡x→∞(2x2+3x+1)\lim_{{x \to \infty}} (2x^2 + 3x + 1)limx→∞​(2x2+3x+1).

Flashcard 30: What is the formal name for lim⁡x→a+f(x)\lim_{{x \to a^+}} f(x)limx→a+​f(x) and lim⁡x→a−f(x)\lim_{{x \to a^-}} f(x)limx→a−​f(x)?

Flashcard 1: How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Flashcard 3: What is the limit of $f(x) = c$ as $x$ approaches any $a$ ?

Flashcard 5: What does it mean if $\lim_{{x \to a}} f(x) = L$ ?

Flashcard 6: Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Flashcard 8: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.

Flashcard 9: How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$ .

Flashcard 13: Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$ .

Flashcard 16: What is the limit notation for $f(x)$ as $x$ approaches infinity?

Flashcard 19: Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$ .

Flashcard 20: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.

Flashcard 21: If $\lim_{{x \to a}} f(x) = L$ , what is the relationship between $f(x)$ and $L$ ?

Flashcard 22: Evaluate $\lim_{{x \to 3}} (x^2 - 9)$ .

Flashcard 23: What does continuity at a point $x = a$ require?

Flashcard 24: What is the limit of $f(x) = x$ as $x$ approaches $a$ ?

Flashcard 25: How is the limit of a function as $x$ approaches $a$ denoted?

Flashcard 28: Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Flashcard 29: Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$ .

Flashcard 30: What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$ ?

Flashcard 1: How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Flashcard 3: What is the limit of $f(x) = c$ as $x$ approaches any $a$ ?

Flashcard 5: What does it mean if $\lim_{{x \to a}} f(x) = L$ ?

Flashcard 6: Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Flashcard 8: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.

Flashcard 9: How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$ .

Flashcard 13: Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$ .

Flashcard 16: What is the limit notation for $f(x)$ as $x$ approaches infinity?

Flashcard 19: Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$ .

Flashcard 20: Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.

Flashcard 21: If $\lim_{{x \to a}} f(x) = L$ , what is the relationship between $f(x)$ and $L$ ?

Flashcard 22: Evaluate $\lim_{{x \to 3}} (x^2 - 9)$ .

Flashcard 23: What does continuity at a point $x = a$ require?

Flashcard 24: What is the limit of $f(x) = x$ as $x$ approaches $a$ ?

Flashcard 25: How is the limit of a function as $x$ approaches $a$ denoted?

Flashcard 28: Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Flashcard 29: Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$ .

Flashcard 30: What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$ ?