Algebraic Properties of Limits - AP Calculus BC
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What is the limit of $f(x) = x \times g(x)$ as $x$ approaches 0 and $g(x)$ is bounded?
What is the limit of $f(x) = x \times g(x)$ as $x$ approaches 0 and $g(x)$ is bounded?
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- If $g(x)$ is bounded, then $x \cdot g(x) \to 0$ as $x \to 0$.
- If $g(x)$ is bounded, then $x \cdot g(x) \to 0$ as $x \to 0$.
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Find the limit of $\frac{3x^2 - x}{2x^2 + x}$ as $x$ approaches infinity.
Find the limit of $\frac{3x^2 - x}{2x^2 + x}$ as $x$ approaches infinity.
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$\frac{3}{2}$. For rational functions, the limit equals the ratio of leading coefficients.
$\frac{3}{2}$. For rational functions, the limit equals the ratio of leading coefficients.
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Evaluate: $\text{lim}_{x \to -2} \frac{x^2 - 4}{x + 2}$.
Evaluate: $\text{lim}_{x \to -2} \frac{x^2 - 4}{x + 2}$.
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-4. Factor as $(x-2)(x+2)/(x+2) = x-2$, then substitute.
-4. Factor as $(x-2)(x+2)/(x+2) = x-2$, then substitute.
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Find the limit: $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$.
Find the limit: $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$.
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- Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.
- Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.
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What is the limit of $\frac{1}{x^2}$ as $x$ approaches infinity?
What is the limit of $\frac{1}{x^2}$ as $x$ approaches infinity?
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- Higher powers in denominators approach zero faster.
- Higher powers in denominators approach zero faster.
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Evaluate: $\text{lim}_{x \to 0} \frac{\tan(x)}{x}$.
Evaluate: $\text{lim}_{x \to 0} \frac{\tan(x)}{x}$.
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- Since $\tan(x) = \sin(x)/\cos(x)$ and $\cos(0) = 1$.
- Since $\tan(x) = \sin(x)/\cos(x)$ and $\cos(0) = 1$.
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Find the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.
Find the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.
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- Factor as $(x-2)(x+2)/(x-2) = x+2$, then substitute.
- Factor as $(x-2)(x+2)/(x-2) = x+2$, then substitute.
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What is the limit of $\frac{1}{x}$ as $x$ approaches infinity?
What is the limit of $\frac{1}{x}$ as $x$ approaches infinity?
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- As $x$ grows large, $\frac{1}{x}$ approaches zero.
- As $x$ grows large, $\frac{1}{x}$ approaches zero.
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Evaluate: $\text{lim}_{x \to 0} \frac{x^3}{x^2}$.
Evaluate: $\text{lim}_{x \to 0} \frac{x^3}{x^2}$.
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- Simplify to $\frac{x^3}{x^2} = x$, then substitute $x=0$.
- Simplify to $\frac{x^3}{x^2} = x$, then substitute $x=0$.
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Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(5x)}{x}$.
Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(5x)}{x}$.
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- Use $\frac{\sin(5x)}{x} = 5 \cdot \frac{\sin(5x)}{5x}$ and the fundamental limit.
- Use $\frac{\sin(5x)}{x} = 5 \cdot \frac{\sin(5x)}{5x}$ and the fundamental limit.
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What is the limit of $x^n$ as $x$ approaches $a$?
What is the limit of $x^n$ as $x$ approaches $a$?
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$a^n$. Power functions are continuous, so direct substitution works.
$a^n$. Power functions are continuous, so direct substitution works.
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Identify the limit: $\text{lim}_{x \to a} c \times f(x)$.
Identify the limit: $\text{lim}_{x \to a} c \times f(x)$.
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$c \times \text{lim}_{x \to a} f(x)$. Constants factor out of limit expressions.
$c \times \text{lim}_{x \to a} f(x)$. Constants factor out of limit expressions.
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Evaluate the limit: $\text{lim}_{x \to -1} (x^2 + 2x + 1)$.
Evaluate the limit: $\text{lim}_{x \to -1} (x^2 + 2x + 1)$.
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- Direct substitution: $(-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$.
- Direct substitution: $(-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$.
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Find the limit: $\text{lim}_{x \to 3}(5x^2 - 2x + 1)$.
Find the limit: $\text{lim}_{x \to 3}(5x^2 - 2x + 1)$.
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- Direct substitution: $5(9) - 2(3) + 1 = 45 - 6 + 1 = 40$.
- Direct substitution: $5(9) - 2(3) + 1 = 45 - 6 + 1 = 40$.
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Find the limit: $\text{lim}_{x \to 2} (5x - 3)$.
Find the limit: $\text{lim}_{x \to 2} (5x - 3)$.
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- Direct substitution: $5(2) - 3 = 10 - 3 = 7$.
- Direct substitution: $5(2) - 3 = 10 - 3 = 7$.
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Evaluate: $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{x}$.
Evaluate: $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{x}$.
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- Use $\frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x}$ and the fundamental limit.
- Use $\frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x}$ and the fundamental limit.
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State the property for the limit of a sum: $\text{lim}_{x \to a} (f(x) + g(x))$.
State the property for the limit of a sum: $\text{lim}_{x \to a} (f(x) + g(x))$.
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$\text{lim}{x \to a} f(x) + \text{lim}{x \to a} g(x)$. The limit of a sum equals the sum of the individual limits.
$\text{lim}{x \to a} f(x) + \text{lim}{x \to a} g(x)$. The limit of a sum equals the sum of the individual limits.
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State the property for the limit of a product: $\text{lim}_{x \to a} (f(x) \times g(x))$.
State the property for the limit of a product: $\text{lim}_{x \to a} (f(x) \times g(x))$.
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$\text{lim}{x \to a} f(x) \times \text{lim}{x \to a} g(x)$. The limit of a product equals the product of individual limits.
$\text{lim}{x \to a} f(x) \times \text{lim}{x \to a} g(x)$. The limit of a product equals the product of individual limits.
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Find the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1.
Find the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1.
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- Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.
- Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.
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Find the limit of a polynomial $f(x)$ as $x$ approaches $a$.
Find the limit of a polynomial $f(x)$ as $x$ approaches $a$.
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$f(a)$. Polynomials are continuous, so use direct substitution.
$f(a)$. Polynomials are continuous, so use direct substitution.
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Evaluate: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.
Evaluate: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.
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- Factor as $(x-3)(x+3)/(x-3) = x+3$, then substitute.
- Factor as $(x-3)(x+3)/(x-3) = x+3$, then substitute.
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Evaluate: $\text{lim}_{x \to 0} \frac{(1+x)^n - 1}{x}$ for $n > 0$.
Evaluate: $\text{lim}_{x \to 0} \frac{(1+x)^n - 1}{x}$ for $n > 0$.
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$n$. This follows from the binomial theorem and L'Hôpital's rule.
$n$. This follows from the binomial theorem and L'Hôpital's rule.
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Evaluate: $\text{lim}_{x \to 1} (x^3 - 3x^2 + 3x - 1)$.
Evaluate: $\text{lim}_{x \to 1} (x^3 - 3x^2 + 3x - 1)$.
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- Direct substitution: $1 - 3 + 3 - 1 = 0$.
- Direct substitution: $1 - 3 + 3 - 1 = 0$.
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What is the limit of $\frac{x^2 - 3x + 2}{x - 2}$ as $x$ approaches 2?
What is the limit of $\frac{x^2 - 3x + 2}{x - 2}$ as $x$ approaches 2?
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- Factor as $ (x-1)(x-2)/(x-2) = x-1 $, then substitute.
- Factor as $ (x-1)(x-2)/(x-2) = x-1 $, then substitute.
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Find the limit: $\text{lim}_{x \to 0} (1 + x)^\frac{1}{x}$.
Find the limit: $\text{lim}_{x \to 0} (1 + x)^\frac{1}{x}$.
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$e$. This is the definition of $e$ as a limit.
$e$. This is the definition of $e$ as a limit.
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State the property for the limit of a difference: $\text{lim}_{x \to a} (f(x) - g(x))$.
State the property for the limit of a difference: $\text{lim}_{x \to a} (f(x) - g(x))$.
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$\text{lim}{x \to a} f(x) - \text{lim}{x \to a} g(x)$. The limit of a difference equals the difference of individual limits.
$\text{lim}{x \to a} f(x) - \text{lim}{x \to a} g(x)$. The limit of a difference equals the difference of individual limits.
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What is the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0?
What is the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0?
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- This is a fundamental trigonometric limit identity.
- This is a fundamental trigonometric limit identity.
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Find the limit: $\text{lim}_{x \to \text{pi}} \text{sin}(x)$.
Find the limit: $\text{lim}_{x \to \text{pi}} \text{sin}(x)$.
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- Direct substitution: $\sin(\pi) = 0$.
- Direct substitution: $\sin(\pi) = 0$.
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What is the limit of $\frac{e^x}{x}$ as $x$ approaches infinity?
What is the limit of $\frac{e^x}{x}$ as $x$ approaches infinity?
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Infinity. Exponential growth dominates polynomial growth.
Infinity. Exponential growth dominates polynomial growth.
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Evaluate: $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$.
Evaluate: $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$.
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- Factor as $(x^2-1)/(x-1) = x+1$, then substitute $x=1$.
- Factor as $(x^2-1)/(x-1) = x+1$, then substitute $x=1$.
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