AP Calculus BC 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 BC.

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: What is the limit of a constant times a function: $\text{lim}_{x \to c} [k \times f(x)]$?

Answer: $k \times \text{lim}_{x \to c} f(x)$. Constants factor out of limits when the limit of the function exists.

Flashcard 2: What is the limit of $x^2$ as $x$ approaches 3?

Answer:

1. Direct substitution works since $x^2$ is continuous at $x = 3$.

Flashcard 3: What is the limit of $\text{cos}(x)$ as $x$ approaches 0?

Answer:

1. Cosine is continuous at zero, so direct substitution gives $\cos(0) = 1$.

Flashcard 4: What is the limit of $x^3$ as $x$ approaches $-2$?

Answer: $-8$. Direct substitution: $(-2)^3 = -8$.

Flashcard 5: State the basic limit property: $\text{lim}_{x \to c} [f(x) \times g(x)]$.

Answer: $\text{lim}_{x \to c} f(x) \times \text{lim}_{x \to c} g(x)$. The limit of a product equals the product of the limits when both limits exist.

Flashcard 6: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

Answer: $\text{lim}_{x \to 0^-} \frac{1}{x} = -\text{inf}$. Approaching zero from negative values makes the fraction arbitrarily large and negative.

Flashcard 7: What does it mean for a limit to be infinite?

Answer: The function grows without bound as $x$ approaches a value. The function increases or decreases without bound near the specified point.

Flashcard 8: What is the limit of $e^{-x}$ as $x$ approaches infinity?

Answer:

1. Exponential decay functions approach zero as the exponent becomes large.

Flashcard 9: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the right?

Answer: $\text{lim}_{x \to 0^+} \frac{1}{x} = +\text{inf}$. Approaching zero from positive values makes the fraction arbitrarily large and positive.

Flashcard 10: What is the limit of $x^2$ as $x$ approaches infinity?

Answer: Infinity. Quadratic functions grow without bound as $x$ approaches infinity.

Flashcard 11: What is the limit of $e^x$ as $x$ approaches infinity?

Answer: Infinity. Exponential functions with base greater than 1 grow without bound as exponent increases.

Flashcard 12: State the limit of a sum of two functions: $\text{lim}_{x \to c} [f(x) + g(x)]$.

Answer: $\text{lim}_{x \to c} f(x) + \text{lim}_{x \to c} g(x)$. The limit of a sum equals the sum of the limits when both limits exist.

Flashcard 13: What is the limit of $x$ as $x$ approaches 5?

Answer:

1. Direct substitution works since $f(x) = x$ is continuous everywhere.

Flashcard 14: What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?

Answer: $-\text{inf}$. The natural logarithm approaches negative infinity as its argument approaches zero.

Flashcard 15: What is the limit of $x^3$ as $x$ approaches infinity?

Answer: Infinity. Cubic functions with positive leading coefficient grow without bound as $x \to \infty$.

Flashcard 16: State the limit property for a function divided by a constant.

Answer: $\frac{\text{lim}_{x \to c} f(x)}{k}$. Dividing by a non-zero constant is equivalent to multiplying by $\frac{1}{k}$.

Flashcard 17: What is the limit of a constant function $f(x) = k$ as $x$ approaches $c$?

Answer: $k$. Constant functions have the same value everywhere, so the limit equals the constant.

Flashcard 18: State the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity.

Answer:

1. The sine function oscillates between -1 and 1, so $\frac{\sin(x)}{x} \to 0$ as $x \to \infty$.

Flashcard 19: State the Squeeze Theorem in limit notation.

Answer: If $g(x) \rightarrow L$ and $h(x) \rightarrow L$, then $f(x) \rightarrow L$. If $f(x)$ is squeezed between two functions with the same limit, $f(x)$ has that limit.

Flashcard 20: What is the limit of $1/x$ as $x$ approaches negative infinity?

Answer:

1. As $x$ becomes large and negative, $\frac{1}{x}$ approaches zero from below.

Flashcard 21: State the condition for the existence of a finite limit of $f(x)$ as $x$ approaches $c$.

Answer: The left-hand and right-hand limits must be equal. This ensures the limit is unique and well-defined.

Flashcard 22: Identify the notation used to denote the right-hand limit.

Answer: $\text{lim}_{x \to c^+} f(x)$. The plus sign indicates approaching from values greater than $c$.

Flashcard 23: Identify the notation used to denote the left-hand limit.

Answer: $\text{lim}_{x \to c^-} f(x)$. The minus sign indicates approaching from values less than $c$.

Flashcard 24: What is the limit of $x^4$ as $x$ approaches 2?

Answer:

1. Direct substitution: $2^4 = 16$.

Flashcard 25: What is the limit of $x^2 + 3x + 2$ as $x$ approaches -1?

Answer:

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

Flashcard 26: State the limit notation for $f(x)$ as $x$ approaches $c$.

Answer: $\text{lim}_{x \to c} f(x)$. Standard notation where $x$ approaches $c$ and we evaluate the limit of $f(x)$.

Flashcard 27: What is the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1?

Answer:

1. Factor and cancel: $\frac{(x-1)(x+1)}{x-1} = x+1$, then substitute $x = 1$.

Flashcard 28: What is the definition of a limit of a function as x approaches a constant c?

Answer: The value that $f(x)$ approaches as $x$ approaches $c$. This describes how a function behaves near a point without requiring the function to be defined there.

Flashcard 29: What is the limit of $\text{tan}(x)$ as $x$ approaches 0?

Answer:

1. Tangent is continuous at zero, so direct substitution gives $\tan(0) = 0$.

Flashcard 30: State the basic limit property: $\text{lim}_{x \to c} [f(x) + g(x)]$.

Answer: $\text{lim}_{x \to c} f(x) + \text{lim}_{x \to c} g(x)$. The limit of a sum equals the sum of the limits when both limits exist.