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AP Calculus BC Flashcards: Lhospitals Rule

Study Lhospitals Rule in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

What this deck covers

This deck focuses on Lhospitals Rule, 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.

AP Calculus BC Flashcards: Lhospitals Rule

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0 reviewed

0% Complete

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QUESTION

Determine the form of $\lim_{x \to \infty} \frac{e^x}{x^3}$ without evaluating.

Tap or drag to reveal answer

ANSWER

$\frac{\infty}{\infty}$ . Both $e^x \to \infty$ and $x^3 \to \infty$ as $x \to \infty$ .

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: Determine the form of $\lim_{x \to \infty} \frac{e^x}{x^3}$ without evaluating.

Answer: $\frac{\infty}{\infty}$ . Both $e^x \to \infty$ and $x^3 \to \infty$ as $x \to \infty$ .

Flashcard 2: What must be true about $f'(x)$ and $g'(x)$ for L'Hospital's Rule to apply?

Answer: $f'(x)$ and $g'(x)$ must exist near $c$ and $g'(x) \neq 0$ . Ensures the rule can be applied and gives a valid result.

Flashcard 3: Evaluate $\lim_{{x \to 0}} \frac{\sin 2x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\sin 2x) = 2\cos 2x$ , so $\frac{2\cos 0}{1} = 2$ .

Flashcard 4: Is L'Hospital's Rule applicable to $\lim_{{x \to 0}} \frac{x^2}{\sin x}$ ?

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $0^2 = 0$ and $\sin 0 = 0$ at $x = 0$ .

Flashcard 5: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\ln x) = \frac{1}{x}$ and $\frac{d}{dx}(x) = 1$ , so $\frac{1/x}{1} \to 0$ .

Flashcard 6: Can L'Hospital's Rule be applied repeatedly?

Answer: Yes, if $\frac{0}{0}$ or $\frac{\infty}{\infty}$ persists after differentiation. Continue applying until a determinate form is reached.

Flashcard 7: Evaluate $\lim_{{x \to \infty}} \frac{2x^2 + 3x}{x^2 - 4}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule: $\frac{4x + 3}{2x} \to \frac{4}{2} = 2$ .

Flashcard 8: Evaluate $\lim_{{x \to 0}} \frac{\arcsin x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ , so $\frac{1}{1} = 1$ .

Flashcard 9: Evaluate $\lim_{{x \to 0}} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\ln(1+x)) = \frac{1}{1+x}$ , so $\frac{1}{1} = 1$ .

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(x) = 1$ and $\frac{d}{dx}(x^2 + 1) = 2x$ , so $\frac{1}{2x} \to 0$ .

Flashcard 11: State L'Hospital's Rule for limits of indeterminate forms.

Answer: $\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)}$ , if the limit exists. Take derivatives of numerator and denominator separately.

Flashcard 12: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x^2}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule: $\frac{1/x}{2x} = \frac{1}{2x^2} \to 0$ .

Flashcard 13: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^3}{e^x}$ .

Answer: Yes, the limit is in the form $\frac{\infty}{\infty}$ . Both $x^3 \to \infty$ and $e^x \to \infty$ as $x \to \infty$ .

Flashcard 14: Find $\lim_{{x \to 0}} \frac{\tan x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\tan x) = \sec^2 x$ and $\frac{d}{dx}(x) = 1$ , so $\frac{1}{1} = 1$ .

Flashcard 15: What is the basic condition to apply L'Hospital's Rule?

Answer: The limit must be in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . These are the only indeterminate forms where L'Hospital's Rule applies.

Flashcard 16: Find $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule twice to get $\frac{2}{e^x} \to 0$ .

Flashcard 17: Determine if L'Hospital's Rule applies: $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $(1)^2 - 1 = 0$ and $1 - 1 = 0$ at $x = 1$ .

Flashcard 18: Find $\lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2}$ using L'Hospital's Rule.

Answer: $\frac{1}{2}$ . Apply L'Hospital's Rule twice: $\frac{e^x - 1}{2x} \to \frac{e^x}{2} = \frac{1}{2}$ .

Flashcard 19: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Answer: $\frac{0}{0}$ . Both $0^3 = 0$ and $e^0 - 1 = 0$ at $x = 0$ .

Flashcard 20: Determine the form for $\lim_{{x \to \infty}} \frac{\ln x}{\sqrt{x}}$ without evaluating.

Answer: $\frac{\infty}{\infty}$ . Both $\ln x \to \infty$ and $\sqrt{x} \to \infty$ as $x \to \infty$ .

Flashcard 21: Evaluate $\lim_{{x \to \infty}} \frac{x}{e^x}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule: $\frac{1}{e^x} \to 0$ as $x \to \infty$ .

Flashcard 22: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $4 - 4 = 0$ and $2 - 2 = 0$ at $x = 2$ .

Flashcard 23: What is the result of $\lim_{{x \to 0}} \frac{e^x - 1}{x}$ using L'Hospital's Rule?

Answer:

$\frac{d}{dx}(e^x - 1) = e^x$ and $\frac{d}{dx}(x) = 1$ , so $\frac{e^0}{1} = 1$ .

Flashcard 24: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ .

Answer: Yes, the limit is in the form $\frac{\infty}{\infty}$ . Both $x^2 \to \infty$ and $e^x \to \infty$ as $x \to \infty$ .

Flashcard 25: Identify the form $\lim_{{x \to 0}} \frac{x}{\sin x}$ without evaluating.

Answer: $\frac{0}{0}$ . Both numerator and denominator approach 0 as $x \to 0$ .

Flashcard 26: Identify if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $\sin 0 = 0$ and $0 = 0$ , giving $\frac{0}{0}$ form.

Flashcard 27: What is the result of $\lim_{{x \to 0}} \frac{1 - \cos x}{x^2}$ using L'Hospital's Rule?

Answer: $\frac{1}{2}$ . Apply L'Hospital's Rule twice: $\frac{\sin x}{2x} \to \frac{\cos x}{2} = \frac{1}{2}$ .

Flashcard 28: Determine if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{x - \sin x}{x^3}$ .

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $0 - \sin 0 = 0$ and $0^3 = 0$ at $x = 0$ .

Flashcard 29: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $4 - 4 = 0$ and $2 - 2 = 0$ at $x = 2$ .

Flashcard 30: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Answer: $\frac{0}{0}$ . Both $0^3 = 0$ and $e^0 - 1 = 0$ at $x = 0$ .

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AP Calculus BC Flashcards: Lhospitals Rule

Study Lhospitals Rule in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

What this deck covers

This deck focuses on Lhospitals Rule, 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.

AP Calculus BC Flashcards: Lhospitals Rule

/ 30

0 reviewed

0% Complete

0 reviewing

QUESTION

Determine the form of $\lim_{x \to \infty} \frac{e^x}{x^3}$ without evaluating.

Tap or drag to reveal answer

ANSWER

$\frac{\infty}{\infty}$ . Both $e^x \to \infty$ and $x^3 \to \infty$ as $x \to \infty$ .

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: Determine the form of $\lim_{x \to \infty} \frac{e^x}{x^3}$ without evaluating.

Answer: $\frac{\infty}{\infty}$ . Both $e^x \to \infty$ and $x^3 \to \infty$ as $x \to \infty$ .

Flashcard 2: What must be true about $f'(x)$ and $g'(x)$ for L'Hospital's Rule to apply?

Answer: $f'(x)$ and $g'(x)$ must exist near $c$ and $g'(x) \neq 0$ . Ensures the rule can be applied and gives a valid result.

Flashcard 3: Evaluate $\lim_{{x \to 0}} \frac{\sin 2x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\sin 2x) = 2\cos 2x$ , so $\frac{2\cos 0}{1} = 2$ .

Flashcard 4: Is L'Hospital's Rule applicable to $\lim_{{x \to 0}} \frac{x^2}{\sin x}$ ?

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $0^2 = 0$ and $\sin 0 = 0$ at $x = 0$ .

Flashcard 5: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\ln x) = \frac{1}{x}$ and $\frac{d}{dx}(x) = 1$ , so $\frac{1/x}{1} \to 0$ .

Flashcard 6: Can L'Hospital's Rule be applied repeatedly?

Answer: Yes, if $\frac{0}{0}$ or $\frac{\infty}{\infty}$ persists after differentiation. Continue applying until a determinate form is reached.

Flashcard 7: Evaluate $\lim_{{x \to \infty}} \frac{2x^2 + 3x}{x^2 - 4}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule: $\frac{4x + 3}{2x} \to \frac{4}{2} = 2$ .

Flashcard 8: Evaluate $\lim_{{x \to 0}} \frac{\arcsin x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ , so $\frac{1}{1} = 1$ .

Flashcard 9: Evaluate $\lim_{{x \to 0}} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\ln(1+x)) = \frac{1}{1+x}$ , so $\frac{1}{1} = 1$ .

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(x) = 1$ and $\frac{d}{dx}(x^2 + 1) = 2x$ , so $\frac{1}{2x} \to 0$ .

Flashcard 11: State L'Hospital's Rule for limits of indeterminate forms.

Answer: $\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)}$ , if the limit exists. Take derivatives of numerator and denominator separately.

Flashcard 12: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x^2}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule: $\frac{1/x}{2x} = \frac{1}{2x^2} \to 0$ .

Flashcard 13: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^3}{e^x}$ .

Answer: Yes, the limit is in the form $\frac{\infty}{\infty}$ . Both $x^3 \to \infty$ and $e^x \to \infty$ as $x \to \infty$ .

Flashcard 14: Find $\lim_{{x \to 0}} \frac{\tan x}{x}$ using L'Hospital's Rule.

Answer:

$\frac{d}{dx}(\tan x) = \sec^2 x$ and $\frac{d}{dx}(x) = 1$ , so $\frac{1}{1} = 1$ .

Flashcard 15: What is the basic condition to apply L'Hospital's Rule?

Answer: The limit must be in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . These are the only indeterminate forms where L'Hospital's Rule applies.

Flashcard 16: Find $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule twice to get $\frac{2}{e^x} \to 0$ .

Flashcard 17: Determine if L'Hospital's Rule applies: $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $(1)^2 - 1 = 0$ and $1 - 1 = 0$ at $x = 1$ .

Flashcard 18: Find $\lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2}$ using L'Hospital's Rule.

Answer: $\frac{1}{2}$ . Apply L'Hospital's Rule twice: $\frac{e^x - 1}{2x} \to \frac{e^x}{2} = \frac{1}{2}$ .

Flashcard 19: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Answer: $\frac{0}{0}$ . Both $0^3 = 0$ and $e^0 - 1 = 0$ at $x = 0$ .

Flashcard 20: Determine the form for $\lim_{{x \to \infty}} \frac{\ln x}{\sqrt{x}}$ without evaluating.

Answer: $\frac{\infty}{\infty}$ . Both $\ln x \to \infty$ and $\sqrt{x} \to \infty$ as $x \to \infty$ .

Flashcard 21: Evaluate $\lim_{{x \to \infty}} \frac{x}{e^x}$ using L'Hospital's Rule.

Answer:

Apply L'Hospital's Rule: $\frac{1}{e^x} \to 0$ as $x \to \infty$ .

Flashcard 22: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $4 - 4 = 0$ and $2 - 2 = 0$ at $x = 2$ .

Flashcard 23: What is the result of $\lim_{{x \to 0}} \frac{e^x - 1}{x}$ using L'Hospital's Rule?

Answer:

$\frac{d}{dx}(e^x - 1) = e^x$ and $\frac{d}{dx}(x) = 1$ , so $\frac{e^0}{1} = 1$ .

Flashcard 24: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ .

Answer: Yes, the limit is in the form $\frac{\infty}{\infty}$ . Both $x^2 \to \infty$ and $e^x \to \infty$ as $x \to \infty$ .

Flashcard 25: Identify the form $\lim_{{x \to 0}} \frac{x}{\sin x}$ without evaluating.

Answer: $\frac{0}{0}$ . Both numerator and denominator approach 0 as $x \to 0$ .

Flashcard 26: Identify if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $\sin 0 = 0$ and $0 = 0$ , giving $\frac{0}{0}$ form.

Flashcard 27: What is the result of $\lim_{{x \to 0}} \frac{1 - \cos x}{x^2}$ using L'Hospital's Rule?

Answer: $\frac{1}{2}$ . Apply L'Hospital's Rule twice: $\frac{\sin x}{2x} \to \frac{\cos x}{2} = \frac{1}{2}$ .

Flashcard 28: Determine if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{x - \sin x}{x^3}$ .

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $0 - \sin 0 = 0$ and $0^3 = 0$ at $x = 0$ .

Flashcard 29: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Answer: Yes, the limit is in the form $\frac{0}{0}$ . Both $4 - 4 = 0$ and $2 - 2 = 0$ at $x = 2$ .

Flashcard 30: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Answer: $\frac{0}{0}$ . Both $0^3 = 0$ and $e^0 - 1 = 0$ at $x = 0$ .

Opening subject page...

AP Calculus BC Flashcards: Lhospitals Rule

What this deck covers

How to use these flashcards

AP Calculus BC Flashcards: Lhospitals Rule

All flashcards

Flashcard 1: Determine the form of lim⁡x→∞exx3\lim_{x \to \infty} \frac{e^x}{x^3}limx→∞​x3ex​ without evaluating.

Flashcard 2: What must be true about f′(x)f'(x)f′(x) and g′(x)g'(x)g′(x) for L'Hospital's Rule to apply?

Flashcard 3: Evaluate lim⁡x→0sin⁡2xx\lim_{{x \to 0}} \frac{\sin 2x}{x}limx→0​xsin2x​ using L'Hospital's Rule.

Flashcard 4: Is L'Hospital's Rule applicable to lim⁡x→0x2sin⁡x\lim_{{x \to 0}} \frac{x^2}{\sin x}limx→0​sinxx2​?

Flashcard 5: Evaluate lim⁡x→∞ln⁡xx\lim_{{x \to \infty}} \frac{\ln x}{x}limx→∞​xlnx​ using L'Hospital's Rule.

Flashcard 6: Can L'Hospital's Rule be applied repeatedly?

Flashcard 7: Evaluate lim⁡x→∞2x2+3xx2−4\lim_{{x \to \infty}} \frac{2x^2 + 3x}{x^2 - 4}limx→∞​x2−42x2+3x​ using L'Hospital's Rule.

Flashcard 8: Evaluate lim⁡x→0arcsin⁡xx\lim_{{x \to 0}} \frac{\arcsin x}{x}limx→0​xarcsinx​ using L'Hospital's Rule.

Flashcard 9: Evaluate lim⁡x→0ln⁡(1+x)x\lim_{{x \to 0}} \frac{\ln(1+x)}{x}limx→0​xln(1+x)​ using L'Hospital's Rule.

Flashcard 10: Evaluate lim⁡x→∞xx2+1\lim_{{x \to \infty}} \frac{x}{x^2 + 1}limx→∞​x2+1x​ using L'Hospital's Rule.

Flashcard 11: State L'Hospital's Rule for limits of indeterminate forms.

Flashcard 12: Evaluate lim⁡x→∞ln⁡xx2\lim_{{x \to \infty}} \frac{\ln x}{x^2}limx→∞​x2lnx​ using L'Hospital's Rule.

Flashcard 13: Determine if L'Hospital's Rule applies: lim⁡x→∞x3ex\lim_{{x \to \infty}} \frac{x^3}{e^x}limx→∞​exx3​.

Flashcard 14: Find lim⁡x→0tan⁡xx\lim_{{x \to 0}} \frac{\tan x}{x}limx→0​xtanx​ using L'Hospital's Rule.

Flashcard 15: What is the basic condition to apply L'Hospital's Rule?

Flashcard 16: Find lim⁡x→∞x2ex\lim_{{x \to \infty}} \frac{x^2}{e^x}limx→∞​exx2​ using L'Hospital's Rule.

Flashcard 17: Determine if L'Hospital's Rule applies: lim⁡x→1x2−1x−1\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}limx→1​x−1x2−1​.

Flashcard 18: Find lim⁡x→0ex−1−xx2\lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2}limx→0​x2ex−1−x​ using L'Hospital's Rule.

Flashcard 19: Determine the form of lim⁡x→0x3ex−1\lim_{{x \to 0}} \frac{x^3}{e^x - 1}limx→0​ex−1x3​ without evaluating.

Flashcard 20: Determine the form for lim⁡x→∞ln⁡xx\lim_{{x \to \infty}} \frac{\ln x}{\sqrt{x}}limx→∞​x​lnx​ without evaluating.

Flashcard 21: Evaluate lim⁡x→∞xex\lim_{{x \to \infty}} \frac{x}{e^x}limx→∞​exx​ using L'Hospital's Rule.

Flashcard 22: Is L'Hospital's Rule applicable to lim⁡x→2x2−4x−2\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}limx→2​x−2x2−4​?

Flashcard 23: What is the result of lim⁡x→0ex−1x\lim_{{x \to 0}} \frac{e^x - 1}{x}limx→0​xex−1​ using L'Hospital's Rule?

Flashcard 24: Determine if L'Hospital's Rule applies: lim⁡x→∞x2ex\lim_{{x \to \infty}} \frac{x^2}{e^x}limx→∞​exx2​.

Flashcard 25: Identify the form lim⁡x→0xsin⁡x\lim_{{x \to 0}} \frac{x}{\sin x}limx→0​sinxx​ without evaluating.

Flashcard 26: Identify if L'Hospital's Rule applies: lim⁡x→0sin⁡xx\lim_{{x \to 0}} \frac{\sin x}{x}limx→0​xsinx​.

Flashcard 27: What is the result of lim⁡x→01−cos⁡xx2\lim_{{x \to 0}} \frac{1 - \cos x}{x^2}limx→0​x21−cosx​ using L'Hospital's Rule?

Flashcard 28: Determine if L'Hospital's Rule applies: lim⁡x→0x−sin⁡xx3\lim_{{x \to 0}} \frac{x - \sin x}{x^3}limx→0​x3x−sinx​.

Flashcard 29: Is L'Hospital's Rule applicable to lim⁡x→2x2−4x−2\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}limx→2​x−2x2−4​?

Flashcard 30: Determine the form of lim⁡x→0x3ex−1\lim_{{x \to 0}} \frac{x^3}{e^x - 1}limx→0​ex−1x3​ without evaluating.

Opening subject page...

AP Calculus BC Flashcards: Lhospitals Rule

What this deck covers

How to use these flashcards

AP Calculus BC Flashcards: Lhospitals Rule

All flashcards

Flashcard 1: Determine the form of lim⁡x→∞exx3\lim_{x \to \infty} \frac{e^x}{x^3}limx→∞​x3ex​ without evaluating.

Flashcard 2: What must be true about f′(x)f'(x)f′(x) and g′(x)g'(x)g′(x) for L'Hospital's Rule to apply?

Flashcard 3: Evaluate lim⁡x→0sin⁡2xx\lim_{{x \to 0}} \frac{\sin 2x}{x}limx→0​xsin2x​ using L'Hospital's Rule.

Flashcard 4: Is L'Hospital's Rule applicable to lim⁡x→0x2sin⁡x\lim_{{x \to 0}} \frac{x^2}{\sin x}limx→0​sinxx2​?

Flashcard 5: Evaluate lim⁡x→∞ln⁡xx\lim_{{x \to \infty}} \frac{\ln x}{x}limx→∞​xlnx​ using L'Hospital's Rule.

Flashcard 6: Can L'Hospital's Rule be applied repeatedly?

Flashcard 7: Evaluate lim⁡x→∞2x2+3xx2−4\lim_{{x \to \infty}} \frac{2x^2 + 3x}{x^2 - 4}limx→∞​x2−42x2+3x​ using L'Hospital's Rule.

Flashcard 8: Evaluate lim⁡x→0arcsin⁡xx\lim_{{x \to 0}} \frac{\arcsin x}{x}limx→0​xarcsinx​ using L'Hospital's Rule.

Flashcard 9: Evaluate lim⁡x→0ln⁡(1+x)x\lim_{{x \to 0}} \frac{\ln(1+x)}{x}limx→0​xln(1+x)​ using L'Hospital's Rule.

Flashcard 10: Evaluate lim⁡x→∞xx2+1\lim_{{x \to \infty}} \frac{x}{x^2 + 1}limx→∞​x2+1x​ using L'Hospital's Rule.

Flashcard 11: State L'Hospital's Rule for limits of indeterminate forms.

Flashcard 12: Evaluate lim⁡x→∞ln⁡xx2\lim_{{x \to \infty}} \frac{\ln x}{x^2}limx→∞​x2lnx​ using L'Hospital's Rule.

Flashcard 13: Determine if L'Hospital's Rule applies: lim⁡x→∞x3ex\lim_{{x \to \infty}} \frac{x^3}{e^x}limx→∞​exx3​.

Flashcard 14: Find lim⁡x→0tan⁡xx\lim_{{x \to 0}} \frac{\tan x}{x}limx→0​xtanx​ using L'Hospital's Rule.

Flashcard 15: What is the basic condition to apply L'Hospital's Rule?

Flashcard 16: Find lim⁡x→∞x2ex\lim_{{x \to \infty}} \frac{x^2}{e^x}limx→∞​exx2​ using L'Hospital's Rule.

Flashcard 17: Determine if L'Hospital's Rule applies: lim⁡x→1x2−1x−1\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}limx→1​x−1x2−1​.

Flashcard 18: Find lim⁡x→0ex−1−xx2\lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2}limx→0​x2ex−1−x​ using L'Hospital's Rule.

Flashcard 19: Determine the form of lim⁡x→0x3ex−1\lim_{{x \to 0}} \frac{x^3}{e^x - 1}limx→0​ex−1x3​ without evaluating.

Flashcard 20: Determine the form for lim⁡x→∞ln⁡xx\lim_{{x \to \infty}} \frac{\ln x}{\sqrt{x}}limx→∞​x​lnx​ without evaluating.

Flashcard 21: Evaluate lim⁡x→∞xex\lim_{{x \to \infty}} \frac{x}{e^x}limx→∞​exx​ using L'Hospital's Rule.

Flashcard 22: Is L'Hospital's Rule applicable to lim⁡x→2x2−4x−2\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}limx→2​x−2x2−4​?

Flashcard 23: What is the result of lim⁡x→0ex−1x\lim_{{x \to 0}} \frac{e^x - 1}{x}limx→0​xex−1​ using L'Hospital's Rule?

Flashcard 24: Determine if L'Hospital's Rule applies: lim⁡x→∞x2ex\lim_{{x \to \infty}} \frac{x^2}{e^x}limx→∞​exx2​.

Flashcard 25: Identify the form lim⁡x→0xsin⁡x\lim_{{x \to 0}} \frac{x}{\sin x}limx→0​sinxx​ without evaluating.

Flashcard 26: Identify if L'Hospital's Rule applies: lim⁡x→0sin⁡xx\lim_{{x \to 0}} \frac{\sin x}{x}limx→0​xsinx​.

Flashcard 27: What is the result of lim⁡x→01−cos⁡xx2\lim_{{x \to 0}} \frac{1 - \cos x}{x^2}limx→0​x21−cosx​ using L'Hospital's Rule?

Flashcard 28: Determine if L'Hospital's Rule applies: lim⁡x→0x−sin⁡xx3\lim_{{x \to 0}} \frac{x - \sin x}{x^3}limx→0​x3x−sinx​.

Flashcard 29: Is L'Hospital's Rule applicable to lim⁡x→2x2−4x−2\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}limx→2​x−2x2−4​?

Flashcard 30: Determine the form of lim⁡x→0x3ex−1\lim_{{x \to 0}} \frac{x^3}{e^x - 1}limx→0​ex−1x3​ without evaluating.

Flashcard 1: Determine the form of $\lim_{x \to \infty} \frac{e^x}{x^3}$ without evaluating.

Flashcard 2: What must be true about $f'(x)$ and $g'(x)$ for L'Hospital's Rule to apply?

Flashcard 3: Evaluate $\lim_{{x \to 0}} \frac{\sin 2x}{x}$ using L'Hospital's Rule.

Flashcard 4: Is L'Hospital's Rule applicable to $\lim_{{x \to 0}} \frac{x^2}{\sin x}$ ?

Flashcard 5: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x}$ using L'Hospital's Rule.

Flashcard 7: Evaluate $\lim_{{x \to \infty}} \frac{2x^2 + 3x}{x^2 - 4}$ using L'Hospital's Rule.

Flashcard 8: Evaluate $\lim_{{x \to 0}} \frac{\arcsin x}{x}$ using L'Hospital's Rule.

Flashcard 9: Evaluate $\lim_{{x \to 0}} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.

Flashcard 12: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x^2}$ using L'Hospital's Rule.

Flashcard 13: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^3}{e^x}$ .

Flashcard 14: Find $\lim_{{x \to 0}} \frac{\tan x}{x}$ using L'Hospital's Rule.

Flashcard 16: Find $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ using L'Hospital's Rule.

Flashcard 17: Determine if L'Hospital's Rule applies: $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Flashcard 18: Find $\lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2}$ using L'Hospital's Rule.

Flashcard 19: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Flashcard 20: Determine the form for $\lim_{{x \to \infty}} \frac{\ln x}{\sqrt{x}}$ without evaluating.

Flashcard 21: Evaluate $\lim_{{x \to \infty}} \frac{x}{e^x}$ using L'Hospital's Rule.

Flashcard 22: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Flashcard 23: What is the result of $\lim_{{x \to 0}} \frac{e^x - 1}{x}$ using L'Hospital's Rule?

Flashcard 24: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ .

Flashcard 25: Identify the form $\lim_{{x \to 0}} \frac{x}{\sin x}$ without evaluating.

Flashcard 26: Identify if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Flashcard 27: What is the result of $\lim_{{x \to 0}} \frac{1 - \cos x}{x^2}$ using L'Hospital's Rule?

Flashcard 28: Determine if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{x - \sin x}{x^3}$ .

Flashcard 29: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Flashcard 30: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Flashcard 1: Determine the form of $\lim_{x \to \infty} \frac{e^x}{x^3}$ without evaluating.

Flashcard 2: What must be true about $f'(x)$ and $g'(x)$ for L'Hospital's Rule to apply?

Flashcard 3: Evaluate $\lim_{{x \to 0}} \frac{\sin 2x}{x}$ using L'Hospital's Rule.

Flashcard 4: Is L'Hospital's Rule applicable to $\lim_{{x \to 0}} \frac{x^2}{\sin x}$ ?

Flashcard 5: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x}$ using L'Hospital's Rule.

Flashcard 7: Evaluate $\lim_{{x \to \infty}} \frac{2x^2 + 3x}{x^2 - 4}$ using L'Hospital's Rule.

Flashcard 8: Evaluate $\lim_{{x \to 0}} \frac{\arcsin x}{x}$ using L'Hospital's Rule.

Flashcard 9: Evaluate $\lim_{{x \to 0}} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.

Flashcard 10: Evaluate $\lim_{{x \to \infty}} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.

Flashcard 12: Evaluate $\lim_{{x \to \infty}} \frac{\ln x}{x^2}$ using L'Hospital's Rule.

Flashcard 13: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^3}{e^x}$ .

Flashcard 14: Find $\lim_{{x \to 0}} \frac{\tan x}{x}$ using L'Hospital's Rule.

Flashcard 16: Find $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ using L'Hospital's Rule.

Flashcard 17: Determine if L'Hospital's Rule applies: $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$ .

Flashcard 18: Find $\lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2}$ using L'Hospital's Rule.

Flashcard 19: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.

Flashcard 20: Determine the form for $\lim_{{x \to \infty}} \frac{\ln x}{\sqrt{x}}$ without evaluating.

Flashcard 21: Evaluate $\lim_{{x \to \infty}} \frac{x}{e^x}$ using L'Hospital's Rule.

Flashcard 22: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Flashcard 23: What is the result of $\lim_{{x \to 0}} \frac{e^x - 1}{x}$ using L'Hospital's Rule?

Flashcard 24: Determine if L'Hospital's Rule applies: $\lim_{{x \to \infty}} \frac{x^2}{e^x}$ .

Flashcard 25: Identify the form $\lim_{{x \to 0}} \frac{x}{\sin x}$ without evaluating.

Flashcard 26: Identify if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{\sin x}{x}$ .

Flashcard 27: What is the result of $\lim_{{x \to 0}} \frac{1 - \cos x}{x^2}$ using L'Hospital's Rule?

Flashcard 28: Determine if L'Hospital's Rule applies: $\lim_{{x \to 0}} \frac{x - \sin x}{x^3}$ .

Flashcard 29: Is L'Hospital's Rule applicable to $\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}$ ?

Flashcard 30: Determine the form of $\lim_{{x \to 0}} \frac{x^3}{e^x - 1}$ without evaluating.