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Title: Limits of Functionss
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Series: Real Analysis
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Chapter: Continuous Functions
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YouTube-Title: Real Analysis 26 | Limits of Functions
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Bright video: https://youtu.be/QoLlvvro6rE
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Dark video: https://youtu.be/RAqbhJ3lnJ0
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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: ra26_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
00:00 Intro
00:59 Definition
05:50 1st Example
06:29 2nd Example (Polynomial)
08:12 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: I \rightarrow \mathbb{R}$ be a function, $x_0 \in I$ and $c \in \mathbb{R}$. What is the correct definition for the notion $$ \lim_{x \rightarrow x_0} f(x) = c$$
A1: For all convergent sequences $(x_n) \subseteq I$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} f(x_n) = c$.
A2: For all convergent sequences $(x_n) \subseteq I\setminus{ x_0 }$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} f(x_n) = c$.
A3: For all convergent sequences $(x_n) \subseteq I\setminus{ x_0 }$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} x_n = c$.
A4: For all convergent sequences $(f(x_n)) \subseteq I\setminus{ x_0 }$ with limit $f(x_0)$, we have $\lim_{n \rightarrow \infty} x_n = c$.
Q2: For the function $ \displaystyle f(x) = \begin{cases} 1 &, ~~ x = 0\ 0 &, ~~x \neq 0\end{cases} $ calculate $$ \lim_{x \rightarrow 0} f(x)$$
A1: $0$
A2: $1$
A3: $2$
A4: It does not exist!
Q3: Which statement is correct for the function $ \displaystyle f(x) = \begin{cases} x^2 &, ~~ x \geq 0 \ 2-x &, ~~x < 0\end{cases} $
A1: $\displaystyle \lim_{x \nearrow 0} f(x) = 0$ and $\displaystyle \lim_{x \searrow 0} f(x) = 0 $ and $\displaystyle \lim_{x \rightarrow 0} f(x) = 0$
A2: $\displaystyle \lim_{x \nearrow 0} f(x) = 0$ and $\displaystyle \lim_{x \searrow 0} f(x) = 1 $ and $\displaystyle \lim_{x \rightarrow 0} f(x) = 2$
A3: $\displaystyle \lim_{x \nearrow 0} f(x) = 2$ and $\displaystyle \lim_{x \searrow 0} f(x) = 0 $ and $\displaystyle \lim_{x \rightarrow 0} f(x)$ does not exist.
A3: $\displaystyle \lim_{x \nearrow 0} f(x)$ and $\displaystyle \lim_{x \searrow 0} f(x)$ and $\displaystyle \lim_{x \rightarrow 0} f(x)$ do not exist.
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Last update: 2025-01
