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: 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 2: 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 3: 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 4: What is the limit of $\text{ln}(x)$ as $x$ approaches Infinity?

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

Flashcard 5: 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 6: 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 7: 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 8: What is the limit of $\text{e}^x - 1$ as $x$ approaches 0?

Answer:

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

Flashcard 9: 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 $\lim f(x) = L$.

Flashcard 10: 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 11: What is the limit of $\text{e}^x - 1$ as $x$ approaches 0?

Answer:

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

Flashcard 12: What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches Infinity?

Answer:

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

Flashcard 13: State the result of evaluating $\text{lim}_{x \to \text{Infinity}} \frac{x^2 + 3x + 2}{5x^2 - 4}$.

Answer: $\frac{1}{5}$. Divide by highest power: coefficients of $x^2$ give $\frac{1}{5}$.

Flashcard 14: 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 15: 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 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: 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 18: 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 19: What is the limit of $\text{sin}(x)$ as $x$ approaches 0?

Answer:

1. Direct substitution: $\sin(0) = 0$.

Flashcard 20: What is the limit of $f(x) = 3x + 5$ as $x$ approaches 2?

Answer:

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

Flashcard 21: 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 22: 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 23: 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 $\lim f(x) = L$.

Flashcard 24: 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 25: 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 26: 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 27: 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 28: 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 29: What is the limit of $\text{ln}(x)$ as $x$ approaches Infinity?

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

Flashcard 30: 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 31: 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 32: 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 33: Identify the procedure for limits with polynomials.

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

Flashcard 34: What is the limit of $\text{cos}(x)$ as $x$ approaches $\frac{\text{pi}}{2}$?

Answer:

1. Direct substitution: $\cos(\frac{\pi}{2}) = 0$.

Flashcard 35: 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 36: Find the limit: $\text{lim}_{x \to 1} (3x^2 + 2x - 1)$.

Answer:

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

Flashcard 37: 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 38: 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 39: 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 40: Which method is used for limits of rational functions with complex roots?

Answer: Factor and Cancel. Factor numerator and denominator, then cancel common factors.

Flashcard 41: 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 42: 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 43: 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 44: State the result of evaluating $\text{lim}_{x \to \text{Infinity}} \frac{x^2 + 3x + 2}{5x^2 - 4}$.

Answer: $\frac{1}{5}$. Divide by highest power: coefficients of $x^2$ give $\frac{1}{5}$.

Flashcard 45: What is the limit of $e^x$ as $x$ approaches 0?

Answer:

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

Flashcard 46: What is the limit of $x^2$ as $x$ approaches -3?

Answer:

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

Flashcard 47: 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 48: What is the limit of $\text{sin}(x)$ as $x$ approaches 0?

Answer:

1. Direct substitution: $\sin(0) = 0$.

Flashcard 49: Evaluate the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$.

Answer:

1. Use $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$: $\frac{\sin(2x)}{x} = 2 \cdot \frac{\sin(2x)}{2x} = 2(1) = 2$.

Flashcard 50: What is the limit of $x^2$ as $x$ approaches -3?

Answer:

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

Flashcard 51: 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 52: 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 53: 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 54: Which method is used for limits of rational functions with complex roots?

Answer: Factor and Cancel. Factor numerator and denominator, then cancel common factors.

Flashcard 55: Evaluate the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$.

Answer:

1. Use $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$: $\frac{\sin(2x)}{x} = 2 \cdot \frac{\sin(2x)}{2x} = 2(1) = 2$.

Flashcard 56: What is the limit of $\text{cos}(x)$ as $x$ approaches $\frac{\text{pi}}{2}$?

Answer:

1. Direct substitution: $\cos(\frac{\pi}{2}) = 0$.

Flashcard 57: 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 58: 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 59: 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 60: 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 61: What is the limit of $e^x$ as $x$ approaches 0?

Answer:

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

Flashcard 62: 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 63: 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 64: 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 65: What is the limit of $f(x) = 3x + 5$ as $x$ approaches 2?

Answer:

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

Flashcard 66: 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 67: Identify the procedure for limits with polynomials.

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

Flashcard 68: 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 69: Find the limit: $\text{lim}_{x \to 1} (3x^2 + 2x - 1)$.

Answer:

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

Flashcard 70: 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 71: 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 72: Find the limit: $\text{lim}_{x \to -\text{Infinity}} (5x^2 + 3x - 2)$.

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