AP Calculus BC Flashcards: Algebraic Properties Of Limits

What this deck covers

This deck focuses on Algebraic Properties Of Limits, 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: Find the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Answer:

1. Factor as $(x-2)(x+2)/(x-2) = x+2$, then substitute.

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

Answer:

1. Direct substitution: $1 - 3 + 3 - 1 = 0$.

Flashcard 3: Find the limit: $\text{lim}_{x \to \text{pi}} \text{sin}(x)$.

Answer:

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

Flashcard 4: What is the limit of $\frac{e^x}{x}$ as $x$ approaches infinity?

Answer: Infinity. Exponential growth dominates polynomial growth.

Flashcard 5: Find the limit of $\frac{3x^2 - x}{2x^2 + x}$ as $x$ approaches infinity.

Answer: $\frac{3}{2}$. For rational functions, the limit equals the ratio of leading coefficients.

Flashcard 6: Evaluate: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Answer:

1. Factor as $(x-3)(x+3)/(x-3) = x+3$, then substitute.

Flashcard 7: Evaluate: $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$.

Answer:

1. Factor as $(x^2-1)/(x-1) = x+1$, then substitute $x=1$.

Flashcard 8: What is the limit of $e^x$ as $x$ approaches 0?

Answer:

1. $e^0 = 1$ by definition of exponential functions.

Flashcard 9: Find the limit: $\text{lim}_{x \to 3}(5x^2 - 2x + 1)$.

Answer:

1. Direct substitution: $5(9) - 2(3) + 1 = 45 - 6 + 1 = 40$.

Flashcard 10: Find the limit of a polynomial $f(x)$ as $x$ approaches $a$.

Answer: $f(a)$. Polynomials are continuous, so use direct substitution.

Flashcard 11: Evaluate: $\text{lim}_{x \to 0} \frac{x^3}{x^2}$.

Answer:

1. Simplify to $\frac{x^3}{x^2} = x$, then substitute $x=0$.

Flashcard 12: Find the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1.

Answer:

1. Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.

Flashcard 13: What is the limit of $x^n$ as $x$ approaches $a$?

Answer: $a^n$. Power functions are continuous, so direct substitution works.

Flashcard 14: Evaluate: $\text{lim}_{x \to 0} (x^4 + 3x^3 - 2x)$.

Answer:

1. Direct substitution: $0 + 0 - 0 = 0$.

Flashcard 15: Evaluate: $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$.

Answer:

1. Factor as $(x^2-1)/(x-1) = x+1$, then substitute $x=1$.

Flashcard 16: Evaluate: $\text{lim}_{x \to 0} \frac{x^3}{x^2}$.

Answer:

1. Simplify to $\frac{x^3}{x^2} = x$, then substitute $x=0$.

Flashcard 17: What is the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0?

Answer:

1. This is a fundamental trigonometric limit identity.

Flashcard 18: Find the limit of $\frac{3x^2 - x}{2x^2 + x}$ as $x$ approaches infinity.

Answer: $\frac{3}{2}$. For rational functions, the limit equals the ratio of leading coefficients.

Flashcard 19: What is the limit of $\frac{1}{x^2}$ as $x$ approaches infinity?

Answer: $0$. Higher powers in denominators approach zero faster.

Flashcard 20: Find the limit: $\text{lim}_{x \to 0} (1 + x)^\frac{1}{x}$.

Answer: $e$. This is the definition of $e$ as a limit.

Flashcard 21: Evaluate: $\lim_{x \to 0} \frac{(1+x)^n - 1}{x}$ for $n > 0$.

Answer: $n$. This follows from the binomial theorem and L'Hôpital's rule.

Flashcard 22: Evaluate: $\text{lim}_{x \to 1} (x^3 - 3x^2 + 3x - 1)$.

Answer:

1. Direct substitution: $1 - 3 + 3 - 1 = 0$.

Flashcard 23: Evaluate: $\text{lim}_{x \to \frac{\text{pi}}{2}} \text{sin}(x)$.

Answer:

1. Direct substitution: $\sin(\pi/2) = 1$.

Flashcard 24: What is the limit of $x^n$ as $x$ approaches $a$?

Answer: $a^n$. Power functions are continuous, so direct substitution works.

Flashcard 25: Evaluate the limit: $\text{lim}_{x \to -1} (x^2 + 2x + 1)$.

Answer:

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

Flashcard 26: What is the limit of $\frac{x^2 - 3x + 2}{x - 2}$ as $x$ approaches 2?

Answer:

1. Factor as $(x-1)(x-2)/(x-2) = x-1$, then substitute.

Flashcard 27: What is the limit of $f(x) = x \times g(x)$ as $x$ approaches 0 and $g(x)$ is bounded?

Answer:

1. If $g(x)$ is bounded, then $x \cdot g(x) \to 0$ as $x \to 0$.

Flashcard 28: Find the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1.

Answer:

1. Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.

Flashcard 29: State the property for the limit of a difference: $\text{lim}_{x \to a} (f(x) - g(x))$.

Answer: $\text{lim}_{x \to a} f(x) - \text{lim}_{x \to a} g(x)$. The limit of a difference equals the difference of individual limits.

Flashcard 30: State the property for the limit of a sum: $\text{lim}_{x \to a} (f(x) + g(x))$.

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

Flashcard 31: Evaluate: $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{x}$.

Answer:

1. Use $\frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x}$ and the fundamental limit.

Flashcard 32: Which property allows splitting limits: $\lim_{x \to a} (f(x) + g(x))$?

Answer: Limit of a sum. This refers to the algebraic property that splits sums.

Flashcard 33: Evaluate: $\text{lim}_{x \to 0} \frac{\tan(x)}{x}$.

Answer:

1. Since $\tan(x) = \sin(x)/\cos(x)$ and $\cos(0) = 1$.

Flashcard 34: What is the limit of $e^x$ as $x$ approaches 0?

Answer:

1. $e^0 = 1$ by definition of exponential functions.

Flashcard 35: Evaluate: $\text{lim}_{x \to -2} \frac{x^2 - 4}{x + 2}$.

Answer: -4. Factor as $(x-2)(x+2)/(x+2) = x-2$, then substitute.

Flashcard 36: Evaluate: $\lim_{x \to -2} \frac{x^2 - 4}{x + 2}$.

Answer: -4. Factor as $(x-2)(x+2)/(x+2) = x-2$, then substitute.

Flashcard 37: Find the limit: $\text{lim}_{x \to 3}(5x^2 - 2x + 1)$.

Answer:

1. Direct substitution: $5(9) - 2(3) + 1 = 45 - 6 + 1 = 40$.

Flashcard 38: State the property for the limit of a quotient: $\lim_{x \to a} \frac{f(x)}{g(x)}$

Answer: $\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$. The limit of a quotient equals the quotient of limits when denominator is nonzero.

Flashcard 39: Find the limit: $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$.

Answer:

1. Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.

Flashcard 40: Find the limit: $\text{lim}_{x \to 2} (5x - 3)$.

Answer:

1. Direct substitution: $5(2) - 3 = 10 - 3 = 7$.

Flashcard 41: Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(5x)}{x}$.

Answer:

1. Use $\frac{\sin(5x)}{x} = 5 \cdot \frac{\sin(5x)}{5x}$ and the fundamental limit.

Flashcard 42: What is the limit of $\frac{1}{x}$ as $x$ approaches infinity?

Answer:

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

Flashcard 43: Identify the limit: $\text{lim}_{x \to a} c \times f(x)$.

Answer: $c \times \text{lim}_{x \to a} f(x)$. Constants factor out of limit expressions.

Flashcard 44: Evaluate: $\text{lim}_{x \to \frac{\text{pi}}{2}} \text{sin}(x)$.

Answer:

1. Direct substitution: $\sin(\pi/2) = 1$.

Flashcard 45: Evaluate: $\lim_{x \to 0} (x^4 + 3x^3 - 2x)$.

Answer:

1. Direct substitution: $0 + 0 - 0 = 0$.

Flashcard 46: Evaluate: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Answer:

1. Factor as $(x-3)(x+3)/(x-3) = x+3$, then substitute.

Flashcard 47: Evaluate: $\text{lim}_{x \to 0} \frac{(1+x)^n - 1}{x}$ for $n > 0$.

Answer: $n$. This follows from the binomial theorem and L'Hôpital's rule.

Flashcard 48: State the property for the limit of a product: $\text{lim}_{x \to a} (f(x) \times g(x))$.

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

Flashcard 49: What is the limit of $\frac{x^2 - 3x + 2}{x - 2}$ as $x$ approaches 2?

Answer:

1. Factor as $(x-1)(x-2)/(x-2) = x-1$, then substitute.

Flashcard 50: State the property for the limit of a quotient: $\text{lim}_{x \to a} \frac{f(x)}{g(x)}$.

Answer: $\frac{\text{lim}_{x \to a} f(x)}{\text{lim}_{x \to a} g(x)}$, if $\text{lim}_{x \to a} g(x) \neq 0$. The limit of a quotient equals the quotient of limits when denominator is nonzero.

Flashcard 51: What is the limit of $\frac{e^x}{x}$ as $x$ approaches infinity?

Answer: Infinity. Exponential growth dominates polynomial growth.

Flashcard 52: Find the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Answer:

1. Factor as $(x-2)(x+2)/(x-2) = x+2$, then substitute.

Flashcard 53: Find the limit: $\text{lim}_{x \to \text{pi}} \text{sin}(x)$.

Answer:

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

Flashcard 54: Which property allows splitting limits: $\lim_{x \to a} (f(x) + g(x))$?

Answer: Limit of a sum. This refers to the algebraic property that splits sums.

Flashcard 55: Find the limit: $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$.

Answer:

1. Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.

Flashcard 56: State the property for the limit of a difference: $\text{lim}_{x \to a} (f(x) - g(x))$.

Answer: $\text{lim}_{x \to a} f(x) - \text{lim}_{x \to a} g(x)$. The limit of a difference equals the difference of individual limits.

Flashcard 57: Evaluate: $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{x}$.

Answer:

1. Use $\frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x}$ and the fundamental limit.

Flashcard 58: What is the limit of a constant $c$ as $x$ approaches any value?

Answer: The limit is $c$. Constants remain unchanged regardless of the variable's value.

Flashcard 59: What is the limit of a constant $c$ as $x$ approaches any value?

Answer: The limit is $c$. Constants remain unchanged regardless of the variable's value.

Flashcard 60: Evaluate the limit: $\text{lim}_{x \to -1} (x^2 + 2x + 1)$.

Answer:

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

Flashcard 61: Identify the limit: $\text{lim}_{x \to a} c \times f(x)$.

Answer: $c \times \text{lim}_{x \to a} f(x)$. Constants factor out of limit expressions.

Flashcard 62: What is the limit of $\frac{1}{x}$ as $x$ approaches infinity?

Answer:

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

Flashcard 63: Find the limit: $\text{lim}_{x \to 2} (5x - 3)$.

Answer:

1. Direct substitution: $5(2) - 3 = 10 - 3 = 7$.

Flashcard 64: Evaluate: $\lim_{x \to 0} \frac{\tan(x)}{x}$.

Answer:

1. Since $\tan(x) = \sin(x)/\cos(x)$ and $\cos(0) = 1$.

Flashcard 65: Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(5x)}{x}$.

Answer:

1. Use $\frac{\sin(5x)}{x} = 5 \cdot \frac{\sin(5x)}{5x}$ and the fundamental limit.

Flashcard 66: What is the limit of $\frac{1}{x^2}$ as $x$ approaches infinity?

Answer:

1. Higher powers in denominators approach zero faster.

Flashcard 67: Find the limit of a polynomial $f(x)$ as $x$ approaches $a$.

Answer: $f(a)$. Polynomials are continuous, so use direct substitution.

Flashcard 68: What is the limit of $\frac{sin(x)}{x}$ as $x$ approaches 0?

Answer:

1. This is a fundamental trigonometric limit identity.

Flashcard 69: State the property for the limit of a product: $\text{lim}_{x \to a} (f(x) \times g(x))$.

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

Flashcard 70: State the property for the limit of a sum: $\text{lim}_{x \to a} (f(x) + g(x))$.

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

Flashcard 71: What is the limit of $f(x) = x \times g(x)$ as $x$ approaches 0 and $g(x)$ is bounded?

Answer:

1. If $g(x)$ is bounded, then $x \cdot g(x) \to 0$ as $x \to 0$.

Flashcard 72: Find the limit: $\text{lim}_{x \to 0} (1 + x)^\frac{1}{x}$.

Answer: $e$. This is the definition of $e$ as a limit.