AP Calculus AB Flashcards: Selecting Procedures For Determining Limits

What this deck covers

This deck focuses on Selecting Procedures For Determining 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: Evaluate the limit: $\lim_{x \to \infty} \frac{2x + 3}{5x - 4}$.

Answer: $\frac{2}{5}$. Divide coefficients of highest powers: $\frac{2}{5}$.

Flashcard 2: What is the limit of $\text{e}^x - 1$ as $x$ approaches 0?

Answer:

1. Direct substitution: $e^0 - 1 = 1 - 1 = 0$.

Flashcard 3: What is the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches Infinity?

Answer:

1. L'Hôpital's Rule: $\lim_{x \to \infty} \frac{\ln(x)}{x} = \lim_{x \to \infty} \frac{1/x}{1} = 0$.

Flashcard 4: What is the limit of $f(x) = 3x + 5$ as $x$ approaches 2?

Answer:

1. Direct substitution: $3(2) + 5 = 11$.

Flashcard 5: Evaluate the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Answer:

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

Flashcard 6: State the definition of continuity at a point.

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

Flashcard 7: What is the limit of $\text{ln}(x)$ as $x$ approaches Infinity?

Answer: Infinity. Logarithm grows without bound as argument increases.

Flashcard 8: Determine the limit: $\text{lim}_{x \to 1} \frac{2x - 2}{x^2 - 1}$.

Answer:

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

Flashcard 9: What is the limit of $f(x) = 2x^3 - x + 5$ as $x$ approaches 0?

Answer:

1. Direct substitution: $2(0)^3 - 0 + 5 = 5$.

Flashcard 10: State the L'Hôpital's Rule.

Answer: If $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$. Apply when limit gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form.

Flashcard 11: What is the limit of $f(x) = \frac{x^2 - 1}{x^2 + 1}$ as $x$ approaches Infinity?

Answer:

1. Divide by highest power: $\frac{x^2}{x^2} = 1$ as $x \to \infty$.

Flashcard 12: Identify the procedure for limits with polynomials.

Answer: Use Direct Substitution if possible. Polynomials are continuous everywhere, so substitute directly.

Flashcard 13: Which theorem states that if $f$ is continuous on $[a, b]$, then $f$ takes every value between $f(a)$ and $f(b)$?

Answer: Intermediate Value Theorem. Guarantees continuous functions achieve all intermediate values.

Flashcard 14: Find the limit: $\text{lim}_{x \to \text{Infinity}} (3 - \frac{5}{x})$.

Answer:

1. As $x \to \infty$, $\frac{5}{x} \to 0$, so limit is $3-0=3$.

Flashcard 15: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

Answer: -Infinity. As $x \to 0^-$, denominator approaches 0 negatively.

Flashcard 16: State the condition for using L'Hôpital's Rule.

Answer: Indeterminate forms $\frac{0}{0}$ or $\frac{\text{Infinity}}{\text{Infinity}}$. Rule applies only to these specific indeterminate forms.

Flashcard 17: What is the limit of $\text{tan}(x)$ as $x$ approaches $\frac{\text{pi}}{4}$?

Answer:

1. Direct substitution: $\tan(\frac{\pi}{4}) = 1$.

Flashcard 18: Find the limit: $\text{lim}_{x \to 1} (3x^2 + 2x - 1)$.

Answer:

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

Flashcard 19: Evaluate the limit: $\text{lim}_{x \to 0} \frac{1 - \text{cos}(x)}{x^2}$.

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

Flashcard 20: State the condition for a function to be continuous at $x = c$.

Answer: $\lim_{x \to c} f(x) = f(c)$. Limit must equal function value for continuity.

Flashcard 21: What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?

Answer: -Infinity. Logarithm approaches $-\infty$ as argument approaches 0.

Flashcard 22: What is the limit of $1/x$ as $x$ approaches 0 from the right?

Answer: Infinity. As $x \to 0^+$, denominator approaches 0 positively.

Flashcard 23: Find the limit: $\text{lim}_{x \to -\text{Infinity}} (5x^2 + 3x - 2)$.

Answer: Infinity. Highest power dominates: $5x^2$ term grows without bound.

Flashcard 24: What is the limit of $e^x$ as $x$ approaches 0?

Answer:

1. Direct substitution: $e^0 = 1$.

Flashcard 25: Determine the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Answer:

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

Flashcard 26: What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$?

Answer:

1. As denominator grows without bound, fraction approaches 0.

Flashcard 27: Evaluate the limit: $\text{lim}_{x \to \text{Infinity}} (\text{e}^{-x})$.

Answer:

1. Exponential decay: $e^{-x} \to 0$ as $x \to \infty$.

Flashcard 28: Which rule applies when directly substituting $x = c$ in rational functions?

Answer: Direct Substitution Rule. When function is continuous at $c$, simply substitute $x=c$.

Flashcard 29: What is the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1?

Answer:

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

Flashcard 30: What is the squeeze theorem used for?

Answer: Determining limits of functions trapped between two other functions. If $g(x) \leq f(x) \leq h(x)$ and $\lim g(x) = \lim h(x) = L$, then $\lim f(x) = L$.