Selecting Procedures for Determining Limits - AP Calculus BC
Card 1 of 30
Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.
Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.
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- Exponential growth dominates logarithmic growth.
- Exponential growth dominates logarithmic growth.
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Which rule is used when both the numerator and denominator approach 0?
Which rule is used when both the numerator and denominator approach 0?
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L'Hôpital's Rule. Applies when limit has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
L'Hôpital's Rule. Applies when limit has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
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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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$-\text{∞}$. Function approaches negative infinity as denominator approaches zero.
$-\text{∞}$. Function approaches negative infinity as denominator approaches zero.
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Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$.
Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$.
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$\frac{5}{2}$. Divide by $x^3$: leading coefficients give $\frac{5}{2}$.
$\frac{5}{2}$. Divide by $x^3$: leading coefficients give $\frac{5}{2}$.
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What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?
What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?
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- This is a fundamental trigonometric limit.
- This is a fundamental trigonometric limit.
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Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$.
Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$.
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- Divide by $x^2$: $1 + \frac{2}{x} + \frac{1}{x^2}$ where terms $\to 0$.
- Divide by $x^2$: $1 + \frac{2}{x} + \frac{1}{x^2}$ where terms $\to 0$.
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Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.
Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.
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- Exponential growth dominates polynomial growth.
- Exponential growth dominates polynomial growth.
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Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.
Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.
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- Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$.
- Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$.
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What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?
What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?
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- Sine oscillates between -1 and 1 while $x$ grows.
- Sine oscillates between -1 and 1 while $x$ grows.
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Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$.
Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$.
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- Use $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$.
- Use $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$.
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What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?
What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?
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$+\text{∞}$. Function approaches positive infinity as denominator approaches zero.
$+\text{∞}$. Function approaches positive infinity as denominator approaches zero.
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What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$?
What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$?
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- Any positive power of zero equals zero.
- Any positive power of zero equals zero.
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Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$.
Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$.
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- This is the derivative of $e^x$ at $x = 0$.
- This is the derivative of $e^x$ at $x = 0$.
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Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$.
Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$.
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- Equivalent to $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.
- Equivalent to $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.
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What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?
What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?
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- Exponential function approaches zero for large negative values.
- Exponential function approaches zero for large negative values.
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Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$.
Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$.
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Does not exist. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.
Does not exist. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.
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What technique is used for limits at infinity involving polynomials?
What technique is used for limits at infinity involving polynomials?
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Divide by the highest power. Standard technique for rational functions at infinity.
Divide by the highest power. Standard technique for rational functions at infinity.
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What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?
What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?
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$\frac{1}{2}$. Standard limit using half-angle trigonometric identity.
$\frac{1}{2}$. Standard limit using half-angle trigonometric identity.
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What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?
What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?
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Divide by the highest power. Divide numerator and denominator by highest degree term.
Divide by the highest power. Divide numerator and denominator by highest degree term.
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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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$-∞$. Natural logarithm approaches negative infinity at zero.
$-∞$. Natural logarithm approaches negative infinity at zero.
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What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?
What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?
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- Exponential growth dominates polynomial growth.
- Exponential growth dominates polynomial growth.
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What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?
What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?
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- Divide by $x$: $\frac{1}{1 + \frac{1}{x}} \to \frac{1}{1} = 1$.
- Divide by $x$: $\frac{1}{1 + \frac{1}{x}} \to \frac{1}{1} = 1$.
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Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$.
Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$.
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- This equals the derivative of $e^x$ at $x = 0$.
- This equals the derivative of $e^x$ at $x = 0$.
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What is the limit of a constant $c$ as $x$ approaches any value?
What is the limit of a constant $c$ as $x$ approaches any value?
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$c$. Constants remain unchanged regardless of variable behavior.
$c$. Constants remain unchanged regardless of variable behavior.
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What is the limit of $e^x$ as $x$ approaches negative infinity?
What is the limit of $e^x$ as $x$ approaches negative infinity?
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- Exponential function approaches zero for large negative values.
- Exponential function approaches zero for large negative values.
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What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?
What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?
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- Standard trigonometric limit equivalent to $\frac{\sin(x)}{x}$.
- Standard trigonometric limit equivalent to $\frac{\sin(x)}{x}$.
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What is the squeeze theorem used for?
What is the squeeze theorem used for?
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Finding limits of bounded functions. Traps function between two converging bounds.
Finding limits of bounded functions. Traps function between two converging bounds.
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Find the limit: $\text{lim}_{x \to \text{∞}} \frac{3x + 2}{2x + 5}$.
Find the limit: $\text{lim}_{x \to \text{∞}} \frac{3x + 2}{2x + 5}$.
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$\frac{3}{2}$. Divide by highest power: $\frac{3}{2} + \frac{\text{terms} \to 0}{\text{terms} \to 0}$.
$\frac{3}{2}$. Divide by highest power: $\frac{3}{2} + \frac{\text{terms} \to 0}{\text{terms} \to 0}$.
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Which method is used for limits of the form $\frac{0}{0}$?
Which method is used for limits of the form $\frac{0}{0}$?
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L'Hôpital's Rule. Standard method for $\frac{0}{0}$ indeterminate forms.
L'Hôpital's Rule. Standard method for $\frac{0}{0}$ indeterminate forms.
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What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?
What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?
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$\frac{1}{2}$. Factor and simplify: $\frac{x-1}{(x+1)(x-1)} = \frac{1}{x+1}$.
$\frac{1}{2}$. Factor and simplify: $\frac{x-1}{(x+1)(x-1)} = \frac{1}{x+1}$.
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