AP Calculus BC Flashcards: Selecting Procedures For Determining Limits

Study Selecting Procedures For Determining Limits in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus BC

Selecting Procedures For Determining Limits

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QUESTION

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What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?

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ANSWER

$-∞$ . Natural logarithm approaches negative infinity at zero.

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What this deck covers

This deck focuses on Selecting Procedures For Determining Limits, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?

Answer: $-∞$ . Natural logarithm approaches negative infinity at zero.

Flashcard 2: Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$ .

Answer:

This is the derivative of $e^x$ at $x = 0$ .

Flashcard 3: What is the squeeze theorem used for?

Answer: Finding limits of bounded functions. Traps function between two converging bounds.

Flashcard 4: What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?

Answer:

Exponential function approaches zero for large negative values.

Flashcard 5: Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$ .

Answer: $\frac{5}{2}$ . Divide by $x^3$ : leading coefficients give $\frac{5}{2}$ .

Flashcard 6: Which method is used for limits of the form $\frac{0}{0}$ ?

Answer: L'Hôpital's Rule. Standard method for $\frac{0}{0}$ indeterminate forms.

Flashcard 7: Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$ .

Answer: $\frac{5}{2}$ . Divide by $x^3$ : leading coefficients give $\frac{5}{2}$ .

Flashcard 8: What is the limit of $(1+\frac{1}{n})^n$ as $n$ approaches infinity?

Answer: $e$ . This is the definition of Euler's number $e$ .

Flashcard 9: What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?

Answer:

Divide by $x$ : $\frac{1}{1 + \frac{1}{x}} \to \frac{1}{1} = 1$ .

Flashcard 10: What is the limit of a constant $c$ as $x$ approaches any value?

Answer: $c$ . Constants remain unchanged regardless of variable behavior.

Flashcard 11: Which rule is used when both the numerator and denominator approach 0?

Answer: L'Hôpital's Rule. Applies when limit has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .

Flashcard 12: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?

Answer: $+\text{∞}$ . Function approaches positive infinity as denominator approaches zero.

Flashcard 13: Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

Answer:

Exponential growth dominates logarithmic growth.

Flashcard 14: What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?

Answer:

Standard trigonometric limit equivalent to $\frac{\sin(x)}{x}$ .

Flashcard 15: Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$ .

Answer:

Divide by $x^2$ : $1 + \frac{2}{x} + \frac{1}{x^2}$ where terms $\to 0$ .

Flashcard 16: What technique is used for limits at infinity involving polynomials?

Answer: Divide by the highest power. Standard technique for rational functions at infinity.

Flashcard 17: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?

Answer: $+\text{∞}$ . Function approaches positive infinity as denominator approaches zero.

Flashcard 18: Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$ .

Answer:

Divide by $x^2$ : $1 + \frac{2}{x} + \frac{1}{x^2}$ where terms $\to 0$ .

Flashcard 19: Find the limit: $\text{lim}_{x \to \text{∞}} \frac{3x + 2}{2x + 5}$ .

Answer: $\frac{3}{2}$ . Divide by highest power: $\frac{3}{2} + \frac{\text{terms} \to 0}{\text{terms} \to 0}$ .

Flashcard 20: Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$ .

Answer:

Use $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$ .

Flashcard 21: What technique is used for limits at infinity involving polynomials?

Answer: Divide by the highest power. Standard technique for rational functions at infinity.

Flashcard 22: Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$ .

Answer:

Equivalent to $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ .

Flashcard 23: What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?

Answer: Divide by the highest power. Divide numerator and denominator by highest degree term.

Flashcard 24: What is the squeeze theorem used for?

Answer: Finding limits of bounded functions. Traps function between two converging bounds.

Flashcard 25: What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

Answer: $\frac{1}{2}$ . Factor and simplify: $\frac{x-1}{(x+1)(x-1)} = \frac{1}{x+1}$ .

Flashcard 26: What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?

Answer:

Exponential growth dominates polynomial growth.

Flashcard 27: What is the limit of $e^x$ as $x$ approaches negative infinity?

Answer:

Exponential function approaches zero for large negative values.

Flashcard 28: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

Answer:

Sine oscillates between $-1$ and $1$ while $x$ grows.

Flashcard 29: What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Answer: $\frac{1}{2}$ . Standard trigonometric limit using half-angle identity.

Flashcard 30: Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.

Answer:

Exponential growth dominates polynomial growth.

Flashcard 31: What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$ ?

Answer:

Any positive power of zero equals zero.

Flashcard 32: What is the limit of a constant $c$ as $x$ approaches any value?

Answer: $c$ . Constants remain unchanged regardless of variable behavior.

Flashcard 33: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

Answer:

Sine oscillates between -1 and 1 while $x$ grows.

Flashcard 34: What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Answer: $\frac{1}{2}$ . Standard limit using half-angle trigonometric identity.

Flashcard 35: What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?

Answer:

Standard trigonometric limit equivalent to $\frac{\sin(x)}{x}$ .

Flashcard 36: What is the limit of $\ln(x)$ as $x$ approaches 0 from the right?

Answer: $-∞$ . Natural logarithm approaches negative infinity at zero.

Flashcard 37: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?

Answer:

This is a fundamental trigonometric limit.

Flashcard 38: What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

Answer: $\frac{1}{2}$ . Factor and cancel: $\frac{x-1}{(x+1)(x-1)} = \frac{1}{x+1}$ .

Flashcard 39: Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$ .

Answer:

This equals the derivative of $e^x$ at $x = 0$ .

Flashcard 40: What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?

Answer:

Exponential growth dominates polynomial growth.

Flashcard 41: Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$ .

Answer:

Equivalent to $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ .

Flashcard 42: Find the limit: $\lim_{x \to \infty} \frac{3x + 2}{2x + 5}$ .

Answer: $\frac{3}{2}$ . Divide by highest power: $\frac{3}{2} + \frac{\text{terms} \to 0}{\text{terms} \to 0}$ .

Flashcard 43: Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$ .

Answer: Does not exist. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$ .

Flashcard 44: Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.

Answer:

Exponential growth dominates polynomial growth.

Flashcard 45: What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?

Answer: Divide by the highest power. Divide numerator and denominator by highest degree term.

Flashcard 46: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

Answer: $-\text{∞}$ . Function approaches negative infinity as denominator approaches zero.

Flashcard 47: Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

Answer:

Exponential growth dominates logarithmic growth.

Flashcard 48: What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Answer: $\frac{1}{2}$ . Standard limit using half-angle trigonometric identity.

Flashcard 49: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?

Answer:

This is a fundamental trigonometric limit.

Flashcard 50: Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$ .

Answer:

Use $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$ .

Flashcard 51: Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

Answer:

Exponential growth dominates logarithmic growth.

Flashcard 52: What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?

Answer:

Exponential function approaches zero for large negative values.

Flashcard 53: Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$ .

Answer:

This equals the derivative of $e^x$ at $x = 0$ .

Flashcard 54: What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?

Answer:

Divide by $x$ : $\frac{1}{1 + \frac{1}{x}} \to \frac{1}{1} = 1$ .

Flashcard 55: Which method is used for limits of the form $\frac{0}{0}$ ?

Answer: L'Hôpital's Rule. Standard method for $\frac{0}{0}$ indeterminate forms.

Flashcard 56: Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$ .

Answer:

This is the derivative of $e^x$ at $x = 0$ .

Flashcard 57: Which rule is used when both the numerator and denominator approach 0?

Answer: L'Hôpital's Rule. Applies when limit has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .

Flashcard 58: Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$ .

Answer:

Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$ .

Flashcard 59: What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

Answer: $\frac{1}{2}$ . Factor and simplify: $\frac{x-1}{(x+1)(x-1)} = \frac{1}{x+1}$ .

Flashcard 60: What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$ ?

Answer:

Any positive power of zero equals zero.

Flashcard 61: Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$ .

Answer: Does not exist. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$ .

Flashcard 62: What is the limit of $(1+\frac{1}{n})^n$ as $n$ approaches infinity?

Answer: $e$ . This is the definition of Euler's number $e$ .

Flashcard 63: What is the limit of $e^x$ as $x$ approaches negative infinity?

Answer:

Exponential function approaches zero for large negative values.

Flashcard 64: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

Answer:

Sine oscillates between -1 and 1 while $x$ grows.

Flashcard 65: Find the limit: $\lim_{x \to 0} \frac{\sin(2x)}{x}$ .

Answer:

Apply $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$ .

Flashcard 66: What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$ ?

Answer:

Any positive power of zero equals zero.

Flashcard 67: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

Answer: $-\text{∞}$ . Function approaches negative infinity as denominator approaches zero.

Flashcard 68: Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$ .

Answer:

Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$ .

AP Calculus BC Flashcards: Selecting Procedures For Determining Limits

Selecting Procedures For Determining Limits

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: What is the limit of ln(x)\text{ln}(x)ln(x) as xxx approaches 0 from the right?

Flashcard 2: Find the limit: limx→0ex−1x\text{lim}_{x \to 0} \frac{e^x - 1}{x}limx→0​xex−1​.

Flashcard 3: What is the squeeze theorem used for?

Flashcard 4: What is the limit of ex\text{e}^xex as xxx approaches negative infinity?

Flashcard 5: Find the limit: limx→∞5x3−x2x3+3x2\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}limx→∞​2x3+3x25x3−x​.

Flashcard 6: Which method is used for limits of the form 00\frac{0}{0}00​?

Flashcard 7: Find the limit: limx→∞5x3−x2x3+3x2\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}limx→∞​2x3+3x25x3−x​.

Flashcard 8: What is the limit of (1+1n)n(1+\frac{1}{n})^n(1+n1​)n as nnn approaches infinity?

Flashcard 9: What is the limit of xx+1\frac{x}{x+1}x+1x​ as xxx approaches infinity?

Flashcard 10: What is the limit of a constant ccc as xxx approaches any value?

Flashcard 11: Which rule is used when both the numerator and denominator approach 0?

Flashcard 12: What is the limit of 1x\frac{1}{x}x1​ as xxx approaches 0 from the positive side?

Flashcard 13: Identify the limit of ln(x)x\frac{\text{ln}(x)}{x}xln(x)​ as xxx approaches infinity.

Flashcard 14: What is the limit of tan(x)x\frac{\text{tan}(x)}{x}xtan(x)​ as xxx approaches 0?

Flashcard 15: Find the limit: limx→−∞x2+2x+1x2\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}limx→−∞​x2x2+2x+1​.

Flashcard 16: What technique is used for limits at infinity involving polynomials?

Flashcard 17: What is the limit of 1x\frac{1}{x}x1​ as xxx approaches 0 from the positive side?

Flashcard 18: Find the limit: limx→−∞x2+2x+1x2\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}limx→−∞​x2x2+2x+1​.

Flashcard 19: Find the limit: limx→∞3x+22x+5\text{lim}_{x \to \text{∞}} \frac{3x + 2}{2x + 5}limx→∞​2x+53x+2​.

Flashcard 20: Find the limit: limx→0sin(2x)x\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}limx→0​xsin(2x)​.

Flashcard 21: What technique is used for limits at infinity involving polynomials?

Flashcard 22: Find the limit: limx→0tan(x)x\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}limx→0​xtan(x)​.

Flashcard 23: What is the method for evaluating limx→∞f(x)\text{lim}_{x \to \text{∞}} f(x)limx→∞​f(x) when f(x)f(x)f(x) is a rational function?

Flashcard 24: What is the squeeze theorem used for?

Flashcard 25: What is the limit of x−1x2−1\frac{x-1}{x^2-1}x2−1x−1​ as xxx approaches 1?

Flashcard 26: What is the limit of x2ex\frac{x^2}{e^x}exx2​ as xxx approaches infinity?

Flashcard 27: What is the limit of exe^xex as xxx approaches negative infinity?

Flashcard 28: What is the limit of sin(x)x\frac{\text{sin}(x)}{x}xsin(x)​ as xxx approaches infinity?

Flashcard 29: What is the limit of 1−cos(x)x2\frac{1 - \text{cos}(x)}{x^2}x21−cos(x)​ as xxx approaches 0?

Flashcard 30: Identify the limit of x3ex\frac{x^3}{\text{e}^x}exx3​ as xxx approaches infinity.

Flashcard 31: What is the limit of xnx^nxn as xxx approaches 0 for n>0n > 0n>0?

Flashcard 32: What is the limit of a constant ccc as xxx approaches any value?

Flashcard 33: What is the limit of sin(x)x\frac{\text{sin}(x)}{x}xsin(x)​ as xxx approaches infinity?

Flashcard 34: What is the limit of 1−cos(x)x2\frac{1 - \text{cos}(x)}{x^2}x21−cos(x)​ as xxx approaches 0?

Flashcard 35: What is the limit of tan(x)x\frac{\text{tan}(x)}{x}xtan(x)​ as xxx approaches 0?

Flashcard 36: What is the limit of ln⁡(x)\ln(x)ln(x) as xxx approaches 0 from the right?

Flashcard 37: What is the limit of sin(x)x\frac{\text{sin}(x)}{x}xsin(x)​ as xxx approaches 0?

Flashcard 38: What is the limit of x−1x2−1\frac{x-1}{x^2-1}x2−1x−1​ as xxx approaches 1?

Flashcard 39: Find the limit: limx→0ex−1x\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}limx→0​xex−1​.

Flashcard 40: What is the limit of x2ex\frac{x^2}{e^x}exx2​ as xxx approaches infinity?

Flashcard 41: Find the limit: limx→0tan(x)x\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}limx→0​xtan(x)​.

Flashcard 42: Find the limit: lim⁡x→∞3x+22x+5\lim_{x \to \infty} \frac{3x + 2}{2x + 5}limx→∞​2x+53x+2​.

Flashcard 43: Find the limit: limx→π2tan(x)\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)limx→2π​​tan(x).

Flashcard 44: Identify the limit of x3ex\frac{x^3}{\text{e}^x}exx3​ as xxx approaches infinity.

Flashcard 45: What is the method for evaluating limx→∞f(x)\text{lim}_{x \to \text{∞}} f(x)limx→∞​f(x) when f(x)f(x)f(x) is a rational function?

Flashcard 46: What is the limit of 1x\frac{1}{x}x1​ as xxx approaches 0 from the left?

Flashcard 47: Identify the limit of ln(x)x\frac{\text{ln}(x)}{x}xln(x)​ as xxx approaches infinity.

Flashcard 48: What is the limit of 1−cos(x)x2\frac{1 - \text{cos}(x)}{x^2}x21−cos(x)​ as xxx approaches 0?

Flashcard 49: What is the limit of sin(x)x\frac{\text{sin}(x)}{x}xsin(x)​ as xxx approaches 0?

Flashcard 50: Find the limit: limx→0sin(2x)x\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}limx→0​xsin(2x)​.

Flashcard 51: Identify the limit of ln(x)x\frac{\text{ln}(x)}{x}xln(x)​ as xxx approaches infinity.

Flashcard 52: What is the limit of ex\text{e}^xex as xxx approaches negative infinity?

Flashcard 53: Find the limit: limx→0ex−1x\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}limx→0​xex−1​.

Flashcard 54: What is the limit of xx+1\frac{x}{x+1}x+1x​ as xxx approaches infinity?

Flashcard 55: Which method is used for limits of the form 00\frac{0}{0}00​?

Flashcard 56: Find the limit: limx→0ex−1x\text{lim}_{x \to 0} \frac{e^x - 1}{x}limx→0​xex−1​.

Flashcard 57: Which rule is used when both the numerator and denominator approach 0?

Flashcard 58: Find the limit: limx→3x2−9x−3\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}limx→3​x−3x2−9​.

Flashcard 59: What is the limit of x−1x2−1\frac{x-1}{x^2-1}x2−1x−1​ as xxx approaches 1?

Flashcard 60: What is the limit of xnx^nxn as xxx approaches 0 for n>0n > 0n>0?

Flashcard 61: Find the limit: limx→π2tan(x)\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)limx→2π​​tan(x).

Flashcard 62: What is the limit of (1+1n)n(1+\frac{1}{n})^n(1+n1​)n as nnn approaches infinity?

Flashcard 63: What is the limit of exe^xex as xxx approaches negative infinity?

Flashcard 64: What is the limit of sin(x)x\frac{\text{sin}(x)}{x}xsin(x)​ as xxx approaches infinity?

Flashcard 65: Find the limit: lim⁡x→0sin⁡(2x)x\lim_{x \to 0} \frac{\sin(2x)}{x}limx→0​xsin(2x)​.

Flashcard 66: What is the limit of xnx^nxn as xxx approaches 0 for n>0n > 0n>0?

Flashcard 67: What is the limit of 1x\frac{1}{x}x1​ as xxx approaches 0 from the left?

Flashcard 68: Find the limit: limx→3x2−9x−3\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}limx→3​x−3x2−9​.

Flashcard 1: What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?

Flashcard 2: Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$ .

Flashcard 4: What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?

Flashcard 5: Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$ .

Flashcard 6: Which method is used for limits of the form $\frac{0}{0}$ ?

Flashcard 7: Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$ .

Flashcard 8: What is the limit of $(1+\frac{1}{n})^n$ as $n$ approaches infinity?

Flashcard 9: What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?

Flashcard 10: What is the limit of a constant $c$ as $x$ approaches any value?

Flashcard 12: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?

Flashcard 13: Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

Flashcard 14: What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?

Flashcard 15: Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$ .

Flashcard 17: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?

Flashcard 18: Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$ .

Flashcard 19: Find the limit: $\text{lim}_{x \to \text{∞}} \frac{3x + 2}{2x + 5}$ .

Flashcard 20: Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$ .

Flashcard 22: Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$ .

Flashcard 23: What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?

Flashcard 25: What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

Flashcard 26: What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?

Flashcard 27: What is the limit of $e^x$ as $x$ approaches negative infinity?

Flashcard 28: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

Flashcard 29: What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Flashcard 30: Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.

Flashcard 31: What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$ ?

Flashcard 32: What is the limit of a constant $c$ as $x$ approaches any value?

Flashcard 33: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

Flashcard 34: What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Flashcard 35: What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?

Flashcard 36: What is the limit of $\ln(x)$ as $x$ approaches 0 from the right?

Flashcard 37: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?

Flashcard 38: What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

Flashcard 39: Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$ .

Flashcard 40: What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?

Flashcard 41: Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$ .

Flashcard 42: Find the limit: $\lim_{x \to \infty} \frac{3x + 2}{2x + 5}$ .

Flashcard 43: Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$ .

Flashcard 44: Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.

Flashcard 45: What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?

Flashcard 46: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

Flashcard 47: Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

Flashcard 48: What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

Flashcard 49: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?

Flashcard 50: Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$ .

Flashcard 51: Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

Flashcard 52: What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?

Flashcard 53: Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$ .

Flashcard 54: What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?

Flashcard 55: Which method is used for limits of the form $\frac{0}{0}$ ?

Flashcard 56: Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$ .

Flashcard 58: Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$ .

Flashcard 59: What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

Flashcard 60: What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$ ?

Flashcard 61: Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$ .

Flashcard 62: What is the limit of $(1+\frac{1}{n})^n$ as $n$ approaches infinity?

Flashcard 63: What is the limit of $e^x$ as $x$ approaches negative infinity?

Flashcard 64: What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

Flashcard 65: Find the limit: $\lim_{x \to 0} \frac{\sin(2x)}{x}$ .

Flashcard 66: What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$ ?

Flashcard 67: What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

Flashcard 68: Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$ .