• Title: Limits of Functionss

  • Series: Real Analysis

  • Chapter: Continuous Functions

  • YouTube-Title: Real Analysis 26 | Limits of Functions

  • Bright video: https://youtu.be/QoLlvvro6rE

  • Dark video: https://youtu.be/RAqbhJ3lnJ0

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ra26_sub_eng.srt missing

  • Timestamps

    00:00 Intro

    00:59 Definition

    05:50 1st Example

    06:29 2nd Example (Polynomial)

    08:12 Credits

  • Subtitle in English (n/a)
  • 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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