Selecting Procedures for Determining Limits - AP Calculus AB
Card 1 of 30
Evaluate the limit: $\lim_{x \to \infty} \frac{2x + 3}{5x - 4}$.
Evaluate the limit: $\lim_{x \to \infty} \frac{2x + 3}{5x - 4}$.
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$\frac{2}{5}$. Divide coefficients of highest powers: $\frac{2}{5}$.
$\frac{2}{5}$. Divide coefficients of highest powers: $\frac{2}{5}$.
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What is the limit of $\text{e}^x - 1$ as $x$ approaches 0?
What is the limit of $\text{e}^x - 1$ as $x$ approaches 0?
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- Direct substitution: $e^0 - 1 = 1 - 1 = 0$.
- Direct substitution: $e^0 - 1 = 1 - 1 = 0$.
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What is the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches Infinity?
What is the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches Infinity?
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- L'Hôpital's Rule: $\lim_{x \to \infty} \frac{\ln(x)}{x} = \lim_{x \to \infty} \frac{1/x}{1} = 0$.
- L'Hôpital's Rule: $\lim_{x \to \infty} \frac{\ln(x)}{x} = \lim_{x \to \infty} \frac{1/x}{1} = 0$.
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What is the limit of $f(x) = 3x + 5$ as $x$ approaches 2?
What is the limit of $f(x) = 3x + 5$ as $x$ approaches 2?
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- Direct substitution: $3(2) + 5 = 11$.
- Direct substitution: $3(2) + 5 = 11$.
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Evaluate the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.
Evaluate the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.
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- Factor: $\frac{(x+3)(x-3)}{x-3} = x+3$, so limit is $3+3=6$.
- Factor: $\frac{(x+3)(x-3)}{x-3} = x+3$, so limit is $3+3=6$.
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State the definition of continuity at a point.
State the definition of continuity at a point.
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A function $f$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. Function is continuous when limit equals function value.
A function $f$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. Function is continuous when limit equals function value.
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What is the limit of $\text{ln}(x)$ as $x$ approaches Infinity?
What is the limit of $\text{ln}(x)$ as $x$ approaches Infinity?
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Infinity. Logarithm grows without bound as argument increases.
Infinity. Logarithm grows without bound as argument increases.
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Determine the limit: $\text{lim}_{x \to 1} \frac{2x - 2}{x^2 - 1}$.
Determine the limit: $\text{lim}_{x \to 1} \frac{2x - 2}{x^2 - 1}$.
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- Factor denominator: $\frac{2(x-1)}{(x+1)(x-1)} = \frac{2}{x+1}$, so limit is $\frac{2}{2}=1$.
- Factor denominator: $\frac{2(x-1)}{(x+1)(x-1)} = \frac{2}{x+1}$, so limit is $\frac{2}{2}=1$.
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What is the limit of $f(x) = 2x^3 - x + 5$ as $x$ approaches 0?
What is the limit of $f(x) = 2x^3 - x + 5$ as $x$ approaches 0?
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- Direct substitution: $2(0)^3 - 0 + 5 = 5$.
- Direct substitution: $2(0)^3 - 0 + 5 = 5$.
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State the L'Hôpital's Rule.
State the L'Hôpital's Rule.
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If $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$. Apply when limit gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form.
If $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$. Apply when limit gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form.
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What is the limit of $f(x) = \frac{x^2 - 1}{x^2 + 1}$ as $x$ approaches Infinity?
What is the limit of $f(x) = \frac{x^2 - 1}{x^2 + 1}$ as $x$ approaches Infinity?
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- Divide by highest power: $\frac{x^2}{x^2} = 1$ as $x \to \infty$.
- Divide by highest power: $\frac{x^2}{x^2} = 1$ as $x \to \infty$.
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Identify the procedure for limits with polynomials.
Identify the procedure for limits with polynomials.
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Use Direct Substitution if possible. Polynomials are continuous everywhere, so substitute directly.
Use Direct Substitution if possible. Polynomials are continuous everywhere, so substitute directly.
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Which theorem states that if $f$ is continuous on $[a, b]$, then $f$ takes every value between $f(a)$ and $f(b)$?
Which theorem states that if $f$ is continuous on $[a, b]$, then $f$ takes every value between $f(a)$ and $f(b)$?
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Intermediate Value Theorem. Guarantees continuous functions achieve all intermediate values.
Intermediate Value Theorem. Guarantees continuous functions achieve all intermediate values.
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Find the limit: $\text{lim}_{x \to \text{Infinity}} (3 - \frac{5}{x})$.
Find the limit: $\text{lim}_{x \to \text{Infinity}} (3 - \frac{5}{x})$.
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- As $x \to \infty$, $\frac{5}{x} \to 0$, so limit is $3-0=3$.
- As $x \to \infty$, $\frac{5}{x} \to 0$, so limit is $3-0=3$.
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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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-Infinity. As $x \to 0^-$, denominator approaches 0 negatively.
-Infinity. As $x \to 0^-$, denominator approaches 0 negatively.
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State the condition for using L'Hôpital's Rule.
State the condition for using L'Hôpital's Rule.
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Indeterminate forms $\frac{0}{0}$ or $\frac{\text{Infinity}}{\text{Infinity}}$. Rule applies only to these specific indeterminate forms.
Indeterminate forms $\frac{0}{0}$ or $\frac{\text{Infinity}}{\text{Infinity}}$. Rule applies only to these specific indeterminate forms.
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What is the limit of $\text{tan}(x)$ as $x$ approaches $\frac{\text{pi}}{4}$?
What is the limit of $\text{tan}(x)$ as $x$ approaches $\frac{\text{pi}}{4}$?
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- Direct substitution: $\tan(\frac{\pi}{4}) = 1$.
- Direct substitution: $\tan(\frac{\pi}{4}) = 1$.
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Find the limit: $\text{lim}_{x \to 1} (3x^2 + 2x - 1)$.
Find the limit: $\text{lim}_{x \to 1} (3x^2 + 2x - 1)$.
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- Direct substitution: $3(1)^2 + 2(1) - 1 = 4$.
- Direct substitution: $3(1)^2 + 2(1) - 1 = 4$.
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Evaluate the limit: $\text{lim}_{x \to 0} \frac{1 - \text{cos}(x)}{x^2}$.
Evaluate the limit: $\text{lim}_{x \to 0} \frac{1 - \text{cos}(x)}{x^2}$.
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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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State the condition for a function to be continuous at $x = c$.
State the condition for a function to be continuous at $x = c$.
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$\lim_{x \to c} f(x) = f(c)$. Limit must equal function value for continuity.
$\lim_{x \to c} f(x) = f(c)$. Limit must equal function value for continuity.
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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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-Infinity. Logarithm approaches $-\infty$ as argument approaches 0.
-Infinity. Logarithm approaches $-\infty$ as argument approaches 0.
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What is the limit of $1/x$ as $x$ approaches 0 from the right?
What is the limit of $1/x$ as $x$ approaches 0 from the right?
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Infinity. As $x \to 0^+$, denominator approaches 0 positively.
Infinity. As $x \to 0^+$, denominator approaches 0 positively.
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Find the limit: $\text{lim}_{x \to -\text{Infinity}} (5x^2 + 3x - 2)$.
Find the limit: $\text{lim}_{x \to -\text{Infinity}} (5x^2 + 3x - 2)$.
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Infinity. Highest power dominates: $5x^2$ term grows without bound.
Infinity. Highest power dominates: $5x^2$ term grows without bound.
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What is the limit of $e^x$ as $x$ approaches 0?
What is the limit of $e^x$ as $x$ approaches 0?
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- Direct substitution: $e^0 = 1$.
- Direct substitution: $e^0 = 1$.
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Determine the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.
Determine the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.
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- Factor: $\frac{(x+2)(x-2)}{x-2} = x+2$, so limit is $2+2=4$.
- Factor: $\frac{(x+2)(x-2)}{x-2} = x+2$, so limit is $2+2=4$.
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What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$?
What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$?
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- As denominator grows without bound, fraction approaches 0.
- As denominator grows without bound, fraction approaches 0.
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Evaluate the limit: $\text{lim}_{x \to \text{Infinity}} (\text{e}^{-x})$.
Evaluate the limit: $\text{lim}_{x \to \text{Infinity}} (\text{e}^{-x})$.
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- Exponential decay: $e^{-x} \to 0$ as $x \to \infty$.
- Exponential decay: $e^{-x} \to 0$ as $x \to \infty$.
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Which rule applies when directly substituting $x = c$ in rational functions?
Which rule applies when directly substituting $x = c$ in rational functions?
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Direct Substitution Rule. When function is continuous at $c$, simply substitute $x=c$.
Direct Substitution Rule. When function is continuous at $c$, simply substitute $x=c$.
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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 numerator: $\frac{(x+1)(x-1)}{x-1} = x+1$, so limit is $1+1=2$.
- Factor numerator: $\frac{(x+1)(x-1)}{x-1} = x+1$, so limit is $1+1=2$.
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What is the squeeze theorem used for?
What is the squeeze theorem used for?
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Determining limits of functions trapped between two other functions. If $g(x) \leq f(x) \leq h(x)$ and $\lim g(x) = \lim h(x) = L$, then $\lim f(x) = L$.
Determining limits of functions trapped between two other functions. If $g(x) \leq f(x) \leq h(x)$ and $\lim g(x) = \lim h(x) = L$, then $\lim f(x) = L$.
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