Defining Limits and Using Limit Notation - AP Calculus BC
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What is the limit of a constant times a function: $\text{lim}_{x \to c} [k \times f(x)]$?
What is the limit of a constant times a function: $\text{lim}_{x \to c} [k \times f(x)]$?
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$k \times \text{lim}_{x \to c} f(x)$. Constants factor out of limits when the limit of the function exists.
$k \times \text{lim}_{x \to c} f(x)$. Constants factor out of limits when the limit of the function exists.
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What is the limit of $x^2$ as $x$ approaches 3?
What is the limit of $x^2$ as $x$ approaches 3?
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- Direct substitution works since $x^2$ is continuous at $x = 3$.
- Direct substitution works since $x^2$ is continuous at $x = 3$.
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What is the limit of $\text{cos}(x)$ as $x$ approaches 0?
What is the limit of $\text{cos}(x)$ as $x$ approaches 0?
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- Cosine is continuous at zero, so direct substitution gives $\cos(0) = 1$.
- Cosine is continuous at zero, so direct substitution gives $\cos(0) = 1$.
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What is the limit of $x^3$ as $x$ approaches $-2$?
What is the limit of $x^3$ as $x$ approaches $-2$?
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$-8$. Direct substitution: $(-2)^3 = -8$.
$-8$. Direct substitution: $(-2)^3 = -8$.
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State the basic limit property: $\text{lim}_{x \to c} [f(x) \times g(x)]$.
State the basic limit property: $\text{lim}_{x \to c} [f(x) \times g(x)]$.
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$\text{lim}{x \to c} f(x) \times \text{lim}{x \to c} g(x)$. The limit of a product equals the product of the limits when both limits exist.
$\text{lim}{x \to c} f(x) \times \text{lim}{x \to c} g(x)$. The limit of a product equals the product of the limits when both limits exist.
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What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?
What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?
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$\text{lim}_{x \to 0^-} \frac{1}{x} = -\text{inf}$. Approaching zero from negative values makes the fraction arbitrarily large and negative.
$\text{lim}_{x \to 0^-} \frac{1}{x} = -\text{inf}$. Approaching zero from negative values makes the fraction arbitrarily large and negative.
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What does it mean for a limit to be infinite?
What does it mean for a limit to be infinite?
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The function grows without bound as $x$ approaches a value. The function increases or decreases without bound near the specified point.
The function grows without bound as $x$ approaches a value. The function increases or decreases without bound near the specified point.
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What is the limit of $e^{-x}$ as $x$ approaches infinity?
What is the limit of $e^{-x}$ as $x$ approaches infinity?
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- Exponential decay functions approach zero as the exponent becomes large.
- Exponential decay functions approach zero as the exponent becomes large.
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What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the right?
What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the right?
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$\text{lim}_{x \to 0^+} \frac{1}{x} = +\text{inf}$. Approaching zero from positive values makes the fraction arbitrarily large and positive.
$\text{lim}_{x \to 0^+} \frac{1}{x} = +\text{inf}$. Approaching zero from positive values makes the fraction arbitrarily large and positive.
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What is the limit of $x^2$ as $x$ approaches infinity?
What is the limit of $x^2$ as $x$ approaches infinity?
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Infinity. Quadratic functions grow without bound as $x$ approaches infinity.
Infinity. Quadratic functions grow without bound as $x$ approaches infinity.
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What is the limit of $e^x$ as $x$ approaches infinity?
What is the limit of $e^x$ as $x$ approaches infinity?
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Infinity. Exponential functions with base greater than 1 grow without bound as exponent increases.
Infinity. Exponential functions with base greater than 1 grow without bound as exponent increases.
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State the limit of a sum of two functions: $\text{lim}_{x \to c} [f(x) + g(x)]$.
State the limit of a sum of two functions: $\text{lim}_{x \to c} [f(x) + g(x)]$.
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$\text{lim}{x \to c} f(x) + \text{lim}{x \to c} g(x)$. The limit of a sum equals the sum of the limits when both limits exist.
$\text{lim}{x \to c} f(x) + \text{lim}{x \to c} g(x)$. The limit of a sum equals the sum of the limits when both limits exist.
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What is the limit of $x$ as $x$ approaches 5?
What is the limit of $x$ as $x$ approaches 5?
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- Direct substitution works since $f(x) = x$ is continuous everywhere.
- Direct substitution works since $f(x) = x$ is continuous everywhere.
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What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?
What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?
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$-\text{inf}$. The natural logarithm approaches negative infinity as its argument approaches zero.
$-\text{inf}$. The natural logarithm approaches negative infinity as its argument approaches zero.
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What is the limit of $x^3$ as $x$ approaches infinity?
What is the limit of $x^3$ as $x$ approaches infinity?
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Infinity. Cubic functions with positive leading coefficient grow without bound as $x \to \infty$.
Infinity. Cubic functions with positive leading coefficient grow without bound as $x \to \infty$.
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State the limit property for a function divided by a constant.
State the limit property for a function divided by a constant.
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$\frac{\text{lim}_{x \to c} f(x)}{k}$. Dividing by a non-zero constant is equivalent to multiplying by $\frac{1}{k}$.
$\frac{\text{lim}_{x \to c} f(x)}{k}$. Dividing by a non-zero constant is equivalent to multiplying by $\frac{1}{k}$.
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What is the limit of a constant function $f(x) = k$ as $x$ approaches $c$?
What is the limit of a constant function $f(x) = k$ as $x$ approaches $c$?
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$k$. Constant functions have the same value everywhere, so the limit equals the constant.
$k$. Constant functions have the same value everywhere, so the limit equals the constant.
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State the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity.
State the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity.
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- The sine function oscillates between -1 and 1, so $\frac{\sin(x)}{x} \to 0$ as $x \to \infty$.
- The sine function oscillates between -1 and 1, so $\frac{\sin(x)}{x} \to 0$ as $x \to \infty$.
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State the Squeeze Theorem in limit notation.
State the Squeeze Theorem in limit notation.
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If $g(x) \rightarrow L$ and $h(x) \rightarrow L$, then $f(x) \rightarrow L$. If $f(x)$ is squeezed between two functions with the same limit, $f(x)$ has that limit.
If $g(x) \rightarrow L$ and $h(x) \rightarrow L$, then $f(x) \rightarrow L$. If $f(x)$ is squeezed between two functions with the same limit, $f(x)$ has that limit.
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What is the limit of $1/x$ as $x$ approaches negative infinity?
What is the limit of $1/x$ as $x$ approaches negative infinity?
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- As $x$ becomes large and negative, $\frac{1}{x}$ approaches zero from below.
- As $x$ becomes large and negative, $\frac{1}{x}$ approaches zero from below.
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State the condition for the existence of a finite limit of $f(x)$ as $x$ approaches $c$.
State the condition for the existence of a finite limit of $f(x)$ as $x$ approaches $c$.
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The left-hand and right-hand limits must be equal. This ensures the limit is unique and well-defined.
The left-hand and right-hand limits must be equal. This ensures the limit is unique and well-defined.
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Identify the notation used to denote the right-hand limit.
Identify the notation used to denote the right-hand limit.
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$\text{lim}_{x \to c^+} f(x)$. The plus sign indicates approaching from values greater than $c$.
$\text{lim}_{x \to c^+} f(x)$. The plus sign indicates approaching from values greater than $c$.
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Identify the notation used to denote the left-hand limit.
Identify the notation used to denote the left-hand limit.
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$\text{lim}_{x \to c^-} f(x)$. The minus sign indicates approaching from values less than $c$.
$\text{lim}_{x \to c^-} f(x)$. The minus sign indicates approaching from values less than $c$.
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What is the limit of $x^4$ as $x$ approaches 2?
What is the limit of $x^4$ as $x$ approaches 2?
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- Direct substitution: $2^4 = 16$.
- Direct substitution: $2^4 = 16$.
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What is the limit of $x^2 + 3x + 2$ as $x$ approaches -1?
What is the limit of $x^2 + 3x + 2$ as $x$ approaches -1?
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- Direct substitution: $(-1)^2 + 3(-1) + 2 = 1 - 3 + 2 = 0$.
- Direct substitution: $(-1)^2 + 3(-1) + 2 = 1 - 3 + 2 = 0$.
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State the limit notation for $f(x)$ as $x$ approaches $c$.
State the limit notation for $f(x)$ as $x$ approaches $c$.
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$\text{lim}_{x \to c} f(x)$. Standard notation where $x$ approaches $c$ and we evaluate the limit of $f(x)$.
$\text{lim}_{x \to c} f(x)$. Standard notation where $x$ approaches $c$ and we evaluate the limit of $f(x)$.
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What is the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1?
What is the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1?
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- Factor and cancel: $\frac{(x-1)(x+1)}{x-1} = x+1$, then substitute $x = 1$.
- Factor and cancel: $\frac{(x-1)(x+1)}{x-1} = x+1$, then substitute $x = 1$.
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What is the definition of a limit of a function as x approaches a constant c?
What is the definition of a limit of a function as x approaches a constant c?
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The value that $f(x)$ approaches as $x$ approaches $c$. This describes how a function behaves near a point without requiring the function to be defined there.
The value that $f(x)$ approaches as $x$ approaches $c$. This describes how a function behaves near a point without requiring the function to be defined there.
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What is the limit of $\text{tan}(x)$ as $x$ approaches 0?
What is the limit of $\text{tan}(x)$ as $x$ approaches 0?
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- Tangent is continuous at zero, so direct substitution gives $\tan(0) = 0$.
- Tangent is continuous at zero, so direct substitution gives $\tan(0) = 0$.
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State the basic limit property: $\text{lim}_{x \to c} [f(x) + g(x)]$.
State the basic limit property: $\text{lim}_{x \to c} [f(x) + g(x)]$.
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$\text{lim}{x \to c} f(x) + \text{lim}{x \to c} g(x)$. The limit of a sum equals the sum of the limits when both limits exist.
$\text{lim}{x \to c} f(x) + \text{lim}{x \to c} g(x)$. The limit of a sum equals the sum of the limits when both limits exist.
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