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Title: Other L’Hospital’s Rules
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Series: Real Analysis
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Chapter: Differentiable Functions
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YouTube-Title: Real Analysis 43 | Other L’Hospital’s Rules
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Bright video: https://youtu.be/KuF0JRsWhBk
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Dark video: https://youtu.be/eI8kObRVSCA
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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: ra43_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 $f,g \colon \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $\lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)}$ makes sense and exists. What is a correct implication using l’Hospital’s rule?
A1: If $\displaystyle \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x)$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.
A2: If $\displaystyle \lim_{x \rightarrow \infty} f(x) \neq \lim_{x \rightarrow \infty} g(x)$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.
A3: If $\displaystyle \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = 0$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.
A4: If $\displaystyle \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = 1$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.
Q2: Let’s apply l’Hospital’s rule for the following limit for $a > 0$: $$ \lim_{x \rightarrow \infty} \frac{ \log(1 + a x) }{ x } $$
A1: $\frac{1}{a}$
A2: $1$
A3: $0$
A4: $a$
A5: $2a$
Q3: Let’s apply l’Hospital’s rule for the following limit: $$ \lim_{x \rightarrow 0} x \log(x) $$
A1: $\frac{1}{2}$
A2: $1$
A3: $0$
A4: $-1$
A5: $\infty$
Q4: Let’s apply l’Hospital’s rule (several times) for the following limit: $$ \lim_{x \rightarrow \infty} \frac{x^5}{\exp(x)} $$
A1: $\frac{1}{2}$
A2: $1$
A3: $0$
A4: $-1$
A5: $\infty$
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Last update: 2025-01