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Title: L’Hospital’s Rule
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Series: Real Analysis
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Chapter: Differentiable Functions
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YouTube-Title: Real Analysis 42 | L’Hospital’s Rule
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Bright video: https://youtu.be/KbS_cRToPFA
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Dark video: https://youtu.be/F3bYc5Syy-o
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra42_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f,g \colon [a,b] \rightarrow \mathbb{R}$ be differentiable functions. What is the correct formulation for the extended mean value theorem?
A1: There is $\hat{x}$ with $f^\prime(\hat{x}) = 0$.
A2: There is $\hat{x}$ with $f^\prime(\hat{x}) = f(a)$.
A3: There is $\hat{x}$ with $\frac{f^\prime(\hat{x})}{g^\prime(\hat{x})} = \frac{f(b) - f(a)}{g(b) - g(a)}$.
A4: There is $\hat{x}$ with $\frac{f^\prime(\hat{x})}{g^\prime(\hat{x})} = (f(b) - f(a) ) \cdot (g(b)-g(a))$.
Q2: Let $f,g \colon [-1,1] \rightarrow \mathbb{R}$ be differentiable functions with $ g^\prime(x) > 0$ for all $x$ and $\lim_{x \rightarrow 0} \frac{f^\prime(x)}{g^\prime(x)}$ exists. What is a correct implication using l’Hospital’s rule?
A1: If $f(0) = g(0)$, then $ \displaystyle \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 0} \frac{f^\prime(x)}{g^\prime(x)} $.
A2: If $f(0) \neq g(0)$, then $ \displaystyle \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 0} \frac{f^\prime(x)}{g^\prime(x)} $.
A3: If $f(0) = g(0) = 0$, then $ \displaystyle \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 0} \frac{f^\prime(x)}{g^\prime(x)} $.
A4: If $f(0) = g(0) = 1$, then $ \displaystyle \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 0} \frac{f^\prime(x)}{g^\prime(x)} $.
Q3: Let’s apply l’Hospital’s rule for the following limit for $a > 0$: $$ \lim_{x \rightarrow 0} \frac{ \log(1 + a x) }{ x } $$
A1: $\frac{1}{a}$
A2: $1$
A3: $0$
A4: $a$
A5: $2a$
Q4: Let’s apply l’Hospital’s rule for the following limit for $a > 0$: $$ \lim_{x \rightarrow 0} \frac{ \sin( a x) }{ a x } $$
A1: $\frac{1}{a}$
A2: $1$
A3: $0$
A4: $a$
A5: $2a$
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Last update: 2025-01
