AP Calculus AB Flashcards: Connecting Multiple Representations Of Limits

What this deck covers

This deck focuses on Connecting Multiple Representations 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.

Flashcard 1: What is the limit of $f(x) = \frac{e^x - 1}{x}$ as $x$ approaches 0?

Answer:

1. Standard exponential limit: derivative of $e^x$ at x=0.

Flashcard 2: Identify the limit: $\lim_{x \to c} 5$ where $c$ is any real number.

Answer: $5$. Constant functions have the same limit everywhere.

Flashcard 3: Identify the limit of $f(x) = x^3$ as $x$ approaches $-1$.

Answer: -1. Direct substitution: $(-1)^3 = -1$.

Flashcard 4: Which theorem guarantees the existence of a limit?

Answer: The Squeeze Theorem. Used when a function is bounded between two functions with the same limit.

Flashcard 5: Identify the limit of $f(x) = \text{tan}(x)$ as $x$ approaches $\frac{\text{π}}{2}$ from the left.

Answer: $+\text{∞}$. Tangent has a vertical asymptote at $\frac{\pi}{2}$, approaching $+\infty$ from the left.

Flashcard 6: State the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the right.

Answer: $+\text{∞}$. As x approaches 0 from positive values, $\frac{1}{x}$ grows without bound.

Flashcard 7: What is the limit of $f(x) = x^2 \text{e}^{-x}$ as $x$ approaches $\text{∞}$?

Answer:

1. Exponential decay dominates polynomial growth at infinity.

Flashcard 8: What is the limit of $f(x) = \frac{\sin(x)}{x}$ as $x$ approaches 0?

Answer:

1. This is a standard trigonometric limit.

Flashcard 9: What is the limit of $f(x) = \frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Answer: $\frac{1}{2}$. Standard limit: $\lim_{x \to 0} \frac{1-\cos(x)}{x^2} = \frac{1}{2}$.

Flashcard 10: Identify the limit: $\lim_{x \to \infty} \frac{1}{x}$.

Answer:

1. As x grows large, $\frac{1}{x}$ approaches 0.

Flashcard 11: State the limit of $f(x) = x^2 \text{sin}(\frac{1}{x})$ as $x$ approaches 0.

Answer:

1. Use Squeeze Theorem: $|\sin(\frac{1}{x})| \leq 1$, so $|x^2\sin(\frac{1}{x})| \leq x^2$.

Flashcard 12: Identify the limit: $\text{lim}_{x \to 0} \frac{\text{cos}(x) - 1}{x}$.

Answer:

1. Use L'Hôpital's rule or the identity $\cos(x) - 1 \approx -\frac{x^2}{2}$.

Flashcard 13: State the limit of $f(x) = \frac{1}{x^2}$ as $x$ approaches 0 from the left.

Answer: $+\infty$. $\frac{1}{x^2}$ is always positive and grows without bound near 0.

Flashcard 14: What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the left?

Answer: $-\text{∞}$. From the left, x is negative, so $\frac{1}{x}$ approaches negative infinity.

Flashcard 15: What is the limit of $f(x) = \frac{x^2 - 4}{x - 2}$ as $x$ approaches 2?

Answer:

1. Factor and cancel: $\frac{(x-2)(x+2)}{x-2} = x+2$, so limit is 4.

Flashcard 16: What is the graphical representation of a limit?

Answer: The $y$-value approaches as $x$ approaches a point. The height the graph approaches as x gets close to the target value.

Flashcard 17: Identify the limit: $\lim_{x \to \infty} (1 + \frac{1}{x})^x$.

Answer: $e$. This is the definition of the mathematical constant e.

Flashcard 18: What is the limit of $f(x) = \frac{x+1}{x-1}$ as $x$ approaches 1?

Answer: Does not exist. The denominator approaches 0 while numerator approaches 2, creating a vertical asymptote.

Flashcard 19: State the limit of $f(x) = \frac{x}{\text{e}^x}$ as $x$ approaches $\text{∞}$.

Answer:

1. Exponential growth dominates linear growth at infinity.

Flashcard 20: What is the limit of $f(x) = \frac{\text{ln}(x+1)}{x}$ as $x$ approaches 0?

Answer:

1. Standard logarithmic limit: derivative of $\ln(x+1)$ at x=0.

Flashcard 21: What is the limit of $f(x) = x \text{e}^{-x}$ as $x$ approaches $\text{∞}$?

Answer:

1. Exponential decay dominates linear growth at infinity.

Flashcard 22: What is the limit of $f(x) = x^2$ as $x$ approaches $3$?

Answer:

1. Direct substitution: $3^2 = 9$.

Flashcard 23: Find the limit: $\lim_{x \to \infty} \frac{2x^3 + 3}{5x^3 + 4}$.

Answer: $\frac{2}{5}$. Divide leading coefficients when degrees are equal: $\frac{2}{5}$.

Flashcard 24: State the condition under which $\lim_{x \to c} f(x)$ exists.

Answer: Left-hand limit equals right-hand limit. Both one-sided limits must exist and be equal.

Flashcard 25: What is the limit of $f(x) = \frac{\sin(2x)}{x}$ as $x$ approaches 0?

Answer:

1. Use $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ with substitution.

Flashcard 26: State the limit of $f(x) = \frac{2x+1}{x+3}$ as $x$ approaches $\text{∞}$.

Answer:

1. Divide leading coefficients: $\frac{2}{1} = 2$.

Flashcard 27: State the limit of $f(x) = \frac{x^3 - 27}{x - 3}$ as $x$ approaches 3.

Answer:

1. Factor: $\frac{(x-3)(x^2+3x+9)}{x-3} = x^2+3x+9$, evaluate at x=3.

Flashcard 28: Identify the limit of $f(x) = 3x + 4$ as $x$ approaches 2.

Answer:

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

Flashcard 29: Find the limit of $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ approaches 3.

Answer:

1. Factor and cancel: $\frac{(x-3)(x+3)}{x-3} = x+3$, so limit is 6.

Flashcard 30: Define a continuous function in terms of limits.

Answer: A function $f$ is continuous at $c$ if $\text{lim}_{x \to c} f(x) = f(c)$. The limit at c equals the function value at c.