AP Calculus AB Flashcards: Defining Limits And Using Limit Notation

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.

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

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

Flashcard 2: 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 3: 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 4: 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 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: What is an asymptote?

Answer: A line that a graph approaches but never touches. Describes boundary behavior where graphs get arbitrarily close but never intersect the line.

Flashcard 7: What does it mean if $\lim_{{x \to a}} f(x)$ does not exist?

Answer: The left-hand and right-hand limits are not equal or at least one is undefined. Different one-sided limits or undefined behavior prevents limit existence.

Flashcard 8: Evaluate $\lim_{{x \to 2}} (3x + 4)$.

Answer:

1. Direct substitution: $3(2) + 4 = 6 + 4 = 10$.

Flashcard 9: 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 10: 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 11: What is an asymptote?

Answer: A line that a graph approaches but never touches. Describes boundary behavior where graphs get arbitrarily close but never intersect the line.

Flashcard 12: 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 13: 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 14: 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 15: 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 16: 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.

Flashcard 17: 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 18: 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 19: Evaluate $\lim_{{x \to 3}} (x^2 - 9)$.

Answer:

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

Flashcard 20: 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 21: Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$.

Answer:

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

Flashcard 22: 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 23: Evaluate $\lim_{{x \to 3}} (x^2 - 9)$.

Answer:

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

Flashcard 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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.

Flashcard 30: 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 31: 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 32: What is a removable discontinuity?

Answer: A discontinuity that can be removed by redefining the function at a point. The limit exists but the function isn't defined or has wrong value at that point.

Flashcard 33: Evaluate $\lim_{x \to \infty} \frac{1}{x}$.

Answer:

1. As $x$ grows without bound, $\frac{1}{x}$ approaches zero.

Flashcard 34: Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$.

Answer:

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

Flashcard 35: What is a removable discontinuity?

Answer: A discontinuity that can be removed by redefining the function at a point. The limit exists but the function isn't defined or has wrong value at that point.

Flashcard 36: 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 37: 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 38: 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 39: What does it mean if $\lim_{{x \to a}} f(x)$ does not exist?

Answer: The left-hand and right-hand limits are not equal or at least one is undefined. Different one-sided limits or undefined behavior prevents limit existence.

Flashcard 40: Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$.

Answer:

1. This is a fundamental trigonometric limit used in calculus.

Flashcard 41: Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$.

Answer:

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

Flashcard 42: What does it mean if $\lim_{{x \to a}} f(x) = \pm \infty$?

Answer: The function $f(x)$ has a vertical asymptote at $x = a$. Function values approach infinity, creating a vertical line the graph approaches.

Flashcard 43: Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$.

Answer:

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

Flashcard 44: 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 45: 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 46: 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 47: Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$.

Answer:

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

Flashcard 48: 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 49: 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 50: Evaluate $\lim_{x \to \infty} \frac{1}{x}$.

Answer:

1. As $x$ grows without bound, $\frac{1}{x}$ approaches zero.

Flashcard 51: 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 52: 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 53: 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 54: 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 55: 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 56: 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 57: 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 58: 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 59: 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 60: Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$.

Answer:

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

Flashcard 61: 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 62: What does it mean if $\lim_{{x \to a}} f(x) = \pm \infty$?

Answer: The function $f(x)$ has a vertical asymptote at $x = a$. Function values approach infinity, creating a vertical line the graph approaches.

Flashcard 63: Evaluate $\lim_{{x \to 2}} (3x + 4)$.

Answer:

1. Direct substitution: $3(2) + 4 = 6 + 4 = 10$.

Flashcard 64: 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 65: 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 66: 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 67: Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$.

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

Flashcard 68: 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 69: Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$.

Answer:

1. This is a fundamental trigonometric limit used in calculus.

Flashcard 70: 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 71: 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 72: 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.