Connecting Multiple Representations of Limits - AP Calculus AB
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What is the limit of $f(x) = \frac{e^x - 1}{x}$ as $x$ approaches 0?
What is the limit of $f(x) = \frac{e^x - 1}{x}$ as $x$ approaches 0?
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- Standard exponential limit: derivative of $e^x$ at x=0.
- Standard exponential limit: derivative of $e^x$ at x=0.
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Identify the limit: $\lim_{x \to c} 5$ where $c$ is any real number.
Identify the limit: $\lim_{x \to c} 5$ where $c$ is any real number.
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$5$. Constant functions have the same limit everywhere.
$5$. Constant functions have the same limit everywhere.
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Identify the limit of $f(x) = x^3$ as $x$ approaches $-1$.
Identify the limit of $f(x) = x^3$ as $x$ approaches $-1$.
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-1. Direct substitution: $(-1)^3 = -1$.
-1. Direct substitution: $(-1)^3 = -1$.
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Which theorem guarantees the existence of a limit?
Which theorem guarantees the existence of a limit?
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The Squeeze Theorem. Used when a function is bounded between two functions with the same limit.
The Squeeze Theorem. Used when a function is bounded between two functions with the same limit.
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Identify the limit of $f(x) = \text{tan}(x)$ as $x$ approaches $\frac{\text{π}}{2}$ from the left.
Identify the limit of $f(x) = \text{tan}(x)$ as $x$ approaches $\frac{\text{π}}{2}$ from the left.
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$+\text{∞}$. Tangent has a vertical asymptote at $\frac{\pi}{2}$, approaching $+\infty$ from the left.
$+\text{∞}$. Tangent has a vertical asymptote at $\frac{\pi}{2}$, approaching $+\infty$ from the left.
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State the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the right.
State the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the right.
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$+\text{∞}$. As x approaches 0 from positive values, $\frac{1}{x}$ grows without bound.
$+\text{∞}$. As x approaches 0 from positive values, $\frac{1}{x}$ grows without bound.
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What is the limit of $f(x) = x^2 \text{e}^{-x}$ as $x$ approaches $\text{∞}$?
What is the limit of $f(x) = x^2 \text{e}^{-x}$ as $x$ approaches $\text{∞}$?
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- Exponential decay dominates polynomial growth at infinity.
- Exponential decay dominates polynomial growth at infinity.
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What is the limit of $f(x) = \frac{\sin(x)}{x}$ as $x$ approaches 0?
What is the limit of $f(x) = \frac{\sin(x)}{x}$ as $x$ approaches 0?
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- This is a standard trigonometric limit.
- This is a standard trigonometric limit.
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What is the limit of $f(x) = \frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?
What is the limit of $f(x) = \frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?
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$\frac{1}{2}$. Standard limit: $\lim_{x \to 0} \frac{1-\cos(x)}{x^2} = \frac{1}{2}$.
$\frac{1}{2}$. Standard limit: $\lim_{x \to 0} \frac{1-\cos(x)}{x^2} = \frac{1}{2}$.
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Identify the limit: $\lim_{x \to \infty} \frac{1}{x}$.
Identify the limit: $\lim_{x \to \infty} \frac{1}{x}$.
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- As x grows large, $\frac{1}{x}$ approaches 0.
- As x grows large, $\frac{1}{x}$ approaches 0.
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State the limit of $f(x) = x^2 \text{sin}(\frac{1}{x})$ as $x$ approaches 0.
State the limit of $f(x) = x^2 \text{sin}(\frac{1}{x})$ as $x$ approaches 0.
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- Use Squeeze Theorem: $|\sin(\frac{1}{x})| \leq 1$, so $|x^2\sin(\frac{1}{x})| \leq x^2$.
- Use Squeeze Theorem: $|\sin(\frac{1}{x})| \leq 1$, so $|x^2\sin(\frac{1}{x})| \leq x^2$.
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Identify the limit: $\text{lim}_{x \to 0} \frac{\text{cos}(x) - 1}{x}$.
Identify the limit: $\text{lim}_{x \to 0} \frac{\text{cos}(x) - 1}{x}$.
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- Use L'Hôpital's rule or the identity $\cos(x) - 1 \approx -\frac{x^2}{2}$.
- Use L'Hôpital's rule or the identity $\cos(x) - 1 \approx -\frac{x^2}{2}$.
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State the limit of $f(x) = \frac{1}{x^2}$ as $x$ approaches 0 from the left.
State the limit of $f(x) = \frac{1}{x^2}$ as $x$ approaches 0 from the left.
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$+\infty$. $\frac{1}{x^2}$ is always positive and grows without bound near 0.
$+\infty$. $\frac{1}{x^2}$ is always positive and grows without bound near 0.
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What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the left?
What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the left?
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$-\text{∞}$. From the left, x is negative, so $\frac{1}{x}$ approaches negative infinity.
$-\text{∞}$. From the left, x is negative, so $\frac{1}{x}$ approaches negative infinity.
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What is the limit of $f(x) = \frac{x^2 - 4}{x - 2}$ as $x$ approaches 2?
What is the limit of $f(x) = \frac{x^2 - 4}{x - 2}$ as $x$ approaches 2?
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- Factor and cancel: $\frac{(x-2)(x+2)}{x-2} = x+2$, so limit is 4.
- Factor and cancel: $\frac{(x-2)(x+2)}{x-2} = x+2$, so limit is 4.
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What is the graphical representation of a limit?
What is the graphical representation of a limit?
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The $y$-value approaches as $x$ approaches a point. The height the graph approaches as x gets close to the target value.
The $y$-value approaches as $x$ approaches a point. The height the graph approaches as x gets close to the target value.
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Identify the limit: $\lim_{x \to \infty} (1 + \frac{1}{x})^x$.
Identify the limit: $\lim_{x \to \infty} (1 + \frac{1}{x})^x$.
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$e$. This is the definition of the mathematical constant e.
$e$. This is the definition of the mathematical constant e.
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What is the limit of $f(x) = \frac{x+1}{x-1}$ as $x$ approaches 1?
What is the limit of $f(x) = \frac{x+1}{x-1}$ as $x$ approaches 1?
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Does not exist. The denominator approaches 0 while numerator approaches 2, creating a vertical asymptote.
Does not exist. The denominator approaches 0 while numerator approaches 2, creating a vertical asymptote.
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State the limit of $f(x) = \frac{x}{\text{e}^x}$ as $x$ approaches $\text{∞}$.
State the limit of $f(x) = \frac{x}{\text{e}^x}$ as $x$ approaches $\text{∞}$.
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- Exponential growth dominates linear growth at infinity.
- Exponential growth dominates linear growth at infinity.
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What is the limit of $f(x) = \frac{\text{ln}(x+1)}{x}$ as $x$ approaches 0?
What is the limit of $f(x) = \frac{\text{ln}(x+1)}{x}$ as $x$ approaches 0?
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- Standard logarithmic limit: derivative of $\ln(x+1)$ at x=0.
- Standard logarithmic limit: derivative of $\ln(x+1)$ at x=0.
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What is the limit of $f(x) = x \text{e}^{-x}$ as $x$ approaches $\text{∞}$?
What is the limit of $f(x) = x \text{e}^{-x}$ as $x$ approaches $\text{∞}$?
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- Exponential decay dominates linear growth at infinity.
- Exponential decay dominates linear growth at infinity.
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What is the limit of $f(x) = x^2$ as $x$ approaches $3$?
What is the limit of $f(x) = x^2$ as $x$ approaches $3$?
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- Direct substitution: $3^2 = 9$.
- Direct substitution: $3^2 = 9$.
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Find the limit: $\lim_{x \to \infty} \frac{2x^3 + 3}{5x^3 + 4}$.
Find the limit: $\lim_{x \to \infty} \frac{2x^3 + 3}{5x^3 + 4}$.
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$\frac{2}{5}$. Divide leading coefficients when degrees are equal: $\frac{2}{5}$.
$\frac{2}{5}$. Divide leading coefficients when degrees are equal: $\frac{2}{5}$.
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State the condition under which $\lim_{x \to c} f(x)$ exists.
State the condition under which $\lim_{x \to c} f(x)$ exists.
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Left-hand limit equals right-hand limit. Both one-sided limits must exist and be equal.
Left-hand limit equals right-hand limit. Both one-sided limits must exist and be equal.
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What is the limit of $f(x) = \frac{\sin(2x)}{x}$ as $x$ approaches 0?
What is the limit of $f(x) = \frac{\sin(2x)}{x}$ as $x$ approaches 0?
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- Use $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ with substitution.
- Use $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ with substitution.
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State the limit of $f(x) = \frac{2x+1}{x+3}$ as $x$ approaches $\text{∞}$.
State the limit of $f(x) = \frac{2x+1}{x+3}$ as $x$ approaches $\text{∞}$.
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- Divide leading coefficients: $\frac{2}{1} = 2$.
- Divide leading coefficients: $\frac{2}{1} = 2$.
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State the limit of $f(x) = \frac{x^3 - 27}{x - 3}$ as $x$ approaches 3.
State the limit of $f(x) = \frac{x^3 - 27}{x - 3}$ as $x$ approaches 3.
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- Factor: $\frac{(x-3)(x^2+3x+9)}{x-3} = x^2+3x+9$, evaluate at x=3.
- Factor: $\frac{(x-3)(x^2+3x+9)}{x-3} = x^2+3x+9$, evaluate at x=3.
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Identify the limit of $f(x) = 3x + 4$ as $x$ approaches 2.
Identify the limit of $f(x) = 3x + 4$ as $x$ approaches 2.
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- Direct substitution: $3(2) + 4 = 10$.
- Direct substitution: $3(2) + 4 = 10$.
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Find the limit of $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ approaches 3.
Find the limit of $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ approaches 3.
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- Factor and cancel: $\frac{(x-3)(x+3)}{x-3} = x+3$, so limit is 6.
- Factor and cancel: $\frac{(x-3)(x+3)}{x-3} = x+3$, so limit is 6.
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Define a continuous function in terms of limits.
Define a continuous function in terms of limits.
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A function $f$ is continuous at $c$ if $\text{lim}_{x \to c} f(x) = f(c)$. The limit at c equals the function value at c.
A function $f$ is continuous at $c$ if $\text{lim}_{x \to c} f(x) = f(c)$. The limit at c equals the function value at c.
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