Connecting Multiple Representations of Limits - AP Calculus BC
Card 1 of 30
Determine the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$.
Determine the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$.
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Does not exist. Tangent has vertical asymptotes where cosine equals zero.
Does not exist. Tangent has vertical asymptotes where cosine equals zero.
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State the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0.
State 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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What is the Squeeze Theorem used for?
What is the Squeeze Theorem used for?
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To find limits of functions squeezed between two functions. Useful when direct evaluation fails but bounds are known.
To find limits of functions squeezed between two functions. Useful when direct evaluation fails but bounds are known.
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Identify the indeterminate form of $\text{∞} - \text{∞}$.
Identify the indeterminate form of $\text{∞} - \text{∞}$.
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Indeterminate form. This form requires algebraic manipulation to resolve the difference.
Indeterminate form. This form requires algebraic manipulation to resolve the difference.
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What is the limit definition of a derivative?
What is the limit definition of a derivative?
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The limit as $h \to 0$ of $\frac{f(x+h) - f(x)}{h}$. This is the formal definition expressing instantaneous rate of change.
The limit as $h \to 0$ of $\frac{f(x+h) - f(x)}{h}$. This is the formal definition expressing instantaneous rate of change.
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Find the limit: $\lim_{x \to \infty} \frac{x^3 - x}{x^3 + x}$.
Find the limit: $\lim_{x \to \infty} \frac{x^3 - x}{x^3 + x}$.
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- Divide numerator and denominator by highest power of $x$.
- Divide numerator and denominator by highest power of $x$.
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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 grow without bound as $x$ increases.
Infinity. Exponential functions grow without bound as $x$ increases.
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State the limit: $\lim_{x \to 0} \frac{\ln(1+x)}{x}$.
State the limit: $\lim_{x \to 0} \frac{\ln(1+x)}{x}$.
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- This is a fundamental logarithmic limit related to derivatives.
- This is a fundamental logarithmic limit related to derivatives.
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What is the limit of $\frac{1 - \cos(x)}{x^2}$ as $x$ approaches 0?
What is the limit of $\frac{1 - \cos(x)}{x^2}$ as $x$ approaches 0?
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$\frac{1}{2}$. This is a fundamental trigonometric limit involving cosine.
$\frac{1}{2}$. This is a fundamental trigonometric limit involving cosine.
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State the limit: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.
State the limit: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.
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- Factor the numerator: $ (x^2-4) = (x-2)(x+2) $, then cancel.
- Factor the numerator: $ (x^2-4) = (x-2)(x+2) $, then cancel.
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Identify the indeterminate form of $\text{∞}^\text{0}$.
Identify the indeterminate form of $\text{∞}^\text{0}$.
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Indeterminate form. This form requires logarithmic techniques to evaluate properly.
Indeterminate form. This form requires logarithmic techniques to evaluate properly.
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Identify the indeterminate form of $\frac{0}{0}$.
Identify the indeterminate form of $\frac{0}{0}$.
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Indeterminate form. This form requires special techniques like L'Hôpital's rule to evaluate.
Indeterminate form. This form requires special techniques like L'Hôpital's rule to evaluate.
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Determine the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(x^2)}{x^2}$.
Determine the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(x^2)}{x^2}$.
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- Let $u = x^2$; as $x \to 0$, $u \to 0$ and use standard limit.
- Let $u = x^2$; as $x \to 0$, $u \to 0$ and use standard limit.
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Find the limit: $\text{lim}_{x \to \text{0}} \frac{\text{tan}(3x)}{x}$.
Find the limit: $\text{lim}_{x \to \text{0}} \frac{\text{tan}(3x)}{x}$.
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- Use substitution with $u = 3x$ and the fundamental limit.
- Use substitution with $u = 3x$ and the fundamental limit.
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Identify the indeterminate form of $0 \times \text{infinity}$.
Identify the indeterminate form of $0 \times \text{infinity}$.
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Indeterminate form. This form requires algebraic manipulation to evaluate properly.
Indeterminate form. This form requires algebraic manipulation to evaluate properly.
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Identify the indeterminate form of $\text{0}^\text{0}$.
Identify the indeterminate form of $\text{0}^\text{0}$.
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Indeterminate form. This form requires logarithmic techniques to evaluate properly.
Indeterminate form. This form requires logarithmic techniques to evaluate properly.
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Identify the indeterminate form of $1^\text{infinity}$.
Identify the indeterminate form of $1^\text{infinity}$.
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Indeterminate form. This form requires logarithmic techniques to evaluate properly.
Indeterminate form. This form requires logarithmic techniques to evaluate properly.
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What is the limit of $\frac{2x}{x+1}$ as $x$ approaches infinity?
What is the limit of $\frac{2x}{x+1}$ as $x$ approaches infinity?
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- Divide numerator and denominator by highest power of $x$.
- Divide numerator and denominator by highest power of $x$.
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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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- Reciprocal functions approach zero as variable grows large.
- Reciprocal functions approach zero as variable grows large.
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Find the limit: $\lim_{x \to \infty} \frac{\ln(x)}{x}$.
Find the limit: $\lim_{x \to \infty} \frac{\ln(x)}{x}$.
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- Logarithmic functions grow slower than any positive power.
- Logarithmic functions grow slower than any positive power.
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Determine the limit: $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$.
Determine the limit: $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$.
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- Factor the numerator: $ (x^2-1) = (x-1)(x+1) $, then cancel.
- Factor the numerator: $ (x^2-1) = (x-1)(x+1) $, then cancel.
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What is the limit of $\frac{\text{sin}(3x)}{x}$ as $x$ approaches 0?
What is the limit of $\frac{\text{sin}(3x)}{x}$ as $x$ approaches 0?
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- Use the identity $\sin(3x) = 3\sin(x)\cos^2(x) - \sin^3(x)$ and limits.
- Use the identity $\sin(3x) = 3\sin(x)\cos^2(x) - \sin^3(x)$ and limits.
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What is the limit of $x^n$ as $x$ approaches 0, where $n > 0$?
What is the limit of $x^n$ as $x$ approaches 0, where $n > 0$?
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- Any positive power of a variable approaching zero gives zero.
- Any positive power of a variable approaching zero gives zero.
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What is the limit of $\text{sin}(x)$ as $x$ approaches infinity?
What is the limit of $\text{sin}(x)$ as $x$ approaches infinity?
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Does not exist. Sine oscillates between -1 and 1, never approaching a single value.
Does not exist. Sine oscillates between -1 and 1, never approaching a single value.
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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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Negative infinity. Natural logarithm approaches negative infinity as input nears zero.
Negative infinity. Natural logarithm approaches negative infinity as input nears zero.
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State L'Hôpital's Rule.
State L'Hôpital's Rule.
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If $\frac{0}{0}$ or $\frac{\text{∞}}{\text{∞}}$, limit is limit of derivatives. Apply when numerator and denominator both approach 0 or infinity.
If $\frac{0}{0}$ or $\frac{\text{∞}}{\text{∞}}$, limit is limit of derivatives. Apply when numerator and denominator both approach 0 or infinity.
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What is the limit of $\text{cos}(x)$ as $x$ approaches infinity?
What is the limit of $\text{cos}(x)$ as $x$ approaches infinity?
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Does not exist. Cosine oscillates between -1 and 1, never approaching a single value.
Does not exist. Cosine oscillates between -1 and 1, never approaching a single value.
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Determine the limit: $\text{lim}_{x \to \text{infinity}} \frac{x}{\text{ln}(x)}$.
Determine the limit: $\text{lim}_{x \to \text{infinity}} \frac{x}{\text{ln}(x)}$.
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Infinity. Linear functions grow faster than logarithmic functions.
Infinity. Linear functions grow faster than logarithmic functions.
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Find the limit: $\text{lim}_{x \to 0} \frac{1 - \text{cos}(x)}{x^2}$.
Find the limit: $\text{lim}_{x \to 0} \frac{1 - \text{cos}(x)}{x^2}$.
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$\frac{1}{2}$. Use the identity $1 - \cos(x) = 2\sin^2(x/2)$ and standard limits.
$\frac{1}{2}$. Use the identity $1 - \cos(x) = 2\sin^2(x/2)$ and standard limits.
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State the limit: $\text{lim}_{x \to \text{infinity}} \frac{5x - 7}{2x + 3}$.
State the limit: $\text{lim}_{x \to \text{infinity}} \frac{5x - 7}{2x + 3}$.
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$\frac{5}{2}$. Divide numerator and denominator by highest power of $x$.
$\frac{5}{2}$. Divide numerator and denominator by highest power of $x$.
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