heureka

Hallo Tali123,

zu b.) Jede Zahl, auch eine negative Zahl ist ein Grenzwert.

Welche der folgenden Grenzwerte existieren ?
Bestimmen sie gegebenfalls den Grenzwert mit der Regel von Bernoulli / de l'Hospital

c.)
\(\displaystyle \lim \limits_{x\to 0} \dfrac{\ln\Big(\cos(x) \Big)} {\ln\Big(\cos(3x)\Big)}\)

Bernoulli / de l'Hospital:

\(\begin{array}{|rcl|} \hline && \mathbf{\lim \limits_{x\to 0} \dfrac{\ln\Big(\cos(x) \Big)} {\ln\Big(\cos(3x)\Big) } } \quad | \quad \text{Bernoulli / de l'Hospital } \\\\ &=& \lim \limits_{x\to 0} \dfrac{\dfrac{-\sin(x)}{\cos(x)}} {\dfrac{-3\sin(3x)}{\cos(3x)} } \\\\ &=& \lim \limits_{x\to 0} \dfrac{1}{3}\cdot \dfrac{\sin(x)\cos(3x)} {\cos(x)\sin(3x)} \\\\ && \begin{array}{|rcl|} \hline \sin(x)\cos(3x) &=& \dfrac12\cdot \Big( \sin(x-3x)+ \sin(x+3x) \Big) \\ &=& \dfrac12\cdot \Big( \sin(-2x)+ \sin(4x) \Big) \\ &=& \dfrac12\cdot \Big( \sin(4x)-\sin(2x) \Big) \\ &=& \dfrac12\cdot \Big( 2\sin(2x)\cos(2x)-\sin(2x) \Big) \\ &=& \dfrac12\cdot\sin(2x) \Big( 2\cos(2x)-1 \Big) \\ \hline \end{array}\\ && \begin{array}{|rcl|} \hline \sin(3x)\cos(x) &=& \dfrac12\cdot \Big( \sin(3x-x)+ \sin(3x+x) \Big) \\ &=& \dfrac12\cdot \Big( \sin(2x)+ \sin(4x) \Big) \\ &=& \dfrac12\cdot \Big( \sin(4x)+\sin(2x) \Big) \\ &=& \dfrac12\cdot\sin(2x) \Big( 2\cos(2x)+1 \Big) \\ \hline \end{array}\\\\ &=& \lim \limits_{x\to 0} \dfrac{1}{3}\cdot \dfrac{\dfrac12\cdot\sin(2x) \Big( 2\cos(2x)-1 \Big)} {\dfrac12\cdot\sin(2x) \Big( 2\cos(2x)+1 \Big)} \\\\ &=& \lim \limits_{x\to 0} \dfrac{1}{3}\cdot \dfrac{\Big( 2\cos(2x)-1 \Big)} {\Big( 2\cos(2x)+1 \Big)} \\\\ &=& \dfrac{1}{3}\cdot \dfrac{\Big( 2\cos(0)-1 \Big)} {\Big( 2\cos(0)+1 \Big)} \quad | \quad \cos(0)=1 \\\\ &=& \dfrac{1}{3}\cdot \dfrac{\Big( 2\cdot 1-1 \Big)} {\Big( 2\cdot 1+1 \Big)} \\\\ &=& \dfrac{1}{3}\cdot \dfrac{ 1 } { 3 } \\\\ &\mathbf{=}& \mathbf{ \dfrac{1}{9} } \\ \hline \end{array}\)

Welche der folgenden Grenzwerte existieren ?
Bestimmen sie gegebenfalls den Grenzwert mit der Regel von Bernoulli / de l'Hospital

b.)
\(\displaystyle \lim \limits_{x\to 0} \dfrac{\cos(x) - 1}{x^2}\)

Bernoulli / de l'Hospital:

\(\begin{array}{|rcll|} \hline && \mathbf{\lim \limits_{x\to 0} \dfrac{\cos(x) - 1}{x^2} } \quad & | \quad \text{1. Bernoulli / de l'Hospital } \\\\ &=& \lim \limits_{x\to 0} \dfrac{-\sin(x)}{2x} \quad & | \quad \text{2. Bernoulli / de l'Hospital } \\\\ &=& \lim \limits_{x\to 0} \dfrac{-\cos(x)}{2} \\\\ &=& \dfrac{-\cos(0)}{2} \quad & | \quad \cos(0)=1 \\\\ &\mathbf{=}& \mathbf{ -\dfrac{1}{2} } \\ \hline \end{array}\)

Welche der folgenden Grenzwerte existieren ?

Bestimmen sie gegebenfalls den Grenzwert mit der Regel von Bernoulli / de l'Hospital

a.)

\(\displaystyle \lim \limits_{x\to 0} \dfrac{\sin(5x)}{\sin(2x)} \)

Bernoulli / de l'Hospital:

\(\begin{array}{|rcll|} \hline && \mathbf{\lim \limits_{x\to 0} \dfrac{\sin(5x)}{\sin(2x)} }\\\\ &=& \lim \limits_{x\to 0} \dfrac{5\cos(5x)}{2\cos(2x)} \\\\ &=& \dfrac{5\cos(0)}{2\cos(0)} \quad & | \quad \cos(0)=1 \\\\ &=& \dfrac{5\cdot 1}{2\cdot 1} \\\\ &\mathbf{=}& \mathbf{ \dfrac{5 }{2 } } \\ \hline \end{array}\)

A regular octagon withside length 4 cm is concentricwith a circle inside,

If the area of the circle is equal to the area to the shaded region between the shapes,

what is the radius of the circle?

\(\text{Let $s =$ octagon side length $ = 4~ cm$}\\ \text{Let $a =$ octagon apothem length}\)

\(\begin{array}{|rcll|} \hline A_{\bigcirc} &=& A_{\text{octagon}} - A_{\bigcirc} \\ 2A_{\bigcirc} &=& A_{\text{octagon}} \\ \mathbf{A_{\bigcirc}} &=& \mathbf{\dfrac{A_{\text{octagon}}} {2}} \\ \hline \end{array}\)

apothem

\(\begin{array}{|rcll|} \hline a &=& \dfrac{s}{2}\cdot \tan\left(90^{\circ}-\frac{45^{\circ}}{2} \right) \\ a &=& \dfrac{s}{2}\cdot \left(1+\sqrt{2} \right) \\\\ A_{\text{octagon}} &=& \dfrac{a \cdot s}{2} \cdot 8 \\ &=& 4as \\ &=& 4 \left( \dfrac{s}{2}\cdot \left(1+\sqrt{2} \right) \right)s \\ &=& 2 s^2 \left(1+\sqrt{2} \right) \\ \mathbf{ \dfrac{A_{\text{octagon}}} {2} } & \mathbf{=} & \mathbf{s^2 \left(1+\sqrt{2} \right)} \\ \hline \end{array} \)

\(\text{Let $A_{\bigcirc} = \pi r^2$ }\)

\(\begin{array}{|rcll|} \hline \mathbf{A_{\bigcirc}} &=& \mathbf{\dfrac{A_{\text{octagon}}} {2}} \\ \pi r^2 &=& s^2 \left(1+\sqrt{2} \right) \\\\ r^2 &=& \dfrac{s^2 \left(1+\sqrt{2} \right)} {\pi} \\\\ r &=& s \sqrt{ \dfrac{ 1+\sqrt{2} } {\pi} } \quad & | \quad s=4~ cm\\\\ r &=& 4 \sqrt{ \dfrac{ 1+\sqrt{2} } {\pi} } \\\\ r &=& 4 \cdot 0.87662309134 \\ r &=& 3.50649236534 \\ \mathbf{r} & \mathbf{=} & \mathbf{3.5~ cm} \\ \hline \end{array}\)

What is the least positive odd integer with exactly nine natural number divisors?

\(\begin{array}{|rcll|} \hline number &=& p^{\color{red}2}q^{\color{red}2} \\ divisors &=& ({\color{red}2}+1)({\color{red}2}+1) = 9 \\ \hline \end{array}\)

The least odd prime numbers are: 3 and 5.

\(\begin{array}{|rcll|} \hline number &=& 3^{\color{red}2}5^{\color{red}2} \\ number &=& 9\cdot 25 \\ \mathbf{number} & \mathbf{=} & \mathbf{225} \\ \hline \end{array} \)

Divisors 225:
1 | 3 | 5 | 9 | 15 | 25 | 45 | 75 | 225 (9 divisors)

In a certain pentagon, the interior angles are   A B C D    and E   where  A B C D E  are integers strictly less than 180 ("Strictly less than 180" means they are "less than and not equal to" 180.)
If the median of the interior angles is  61  and there is only one mode,

then what are the degree measures of all five angles?

Given that  \(\large{x}\)  is a positive integer less than 100,
how many solutions does the congruence  have?
\(\large{x + 13 \equiv 55 \pmod{34}}\)

\(\begin{array}{|rcll|} \hline x + 13 &\equiv& 55 \pmod{34} \\ x + 13 &\equiv& 55-34 \pmod{34} \\ x + 13 &\equiv& 21 \pmod{34} \quad & | \quad - 13 \\ x &\equiv& 21-13 \pmod{34} \\ x &\equiv& 8 \pmod{34} \\ \mathbf{x} & \mathbf{=} & \mathbf{8 + n \cdot 34}, ~ n \in \mathbb{N} \\ \hline \end{array} \)

\(\begin{array}{|c|l|c|} \hline n, ~ n \in \mathbb{N} & \mathbf{x = 8 + n \cdot 34} & ~ x>0,~ x<100 \\ \hline 0 & x = 8+0\cdot 34 \\ & x= 8 & \checkmark \\ \hline 1 & x = 8+1\cdot 34 \\ & x= 42 & \checkmark \\ \hline 2 & x = 8+2\cdot34 \\ & x = 76 & \checkmark \\ \hline 3 & x = 8+3\cdot 34 \\ & x = 110 & x> 100 \\ \hline \ldots & & x> 100 \\ \hline \end{array}\)

The congruence  has three solutions: \(\mathbf{x = 8}\) and \(\mathbf{x = 42}\) and \(\mathbf{x = 76}\)

Hallo Tali123,

ich denke mehr wüssen wir nicht rechnen, denn wir haben bewiesen, das die Induktionsbehauptung mit dem Induktionsschluss übereinstimmt.

the positive integers are arranged in the pattern illustrated below if this pattern continues indefinitely, what is the number immediately above 39863?

1  2 5 10 17 26

3  4 7 12 19 28

6  8 9 14 21 30

11 13 15 16 23 32

18 20 22 24 25 34

27 29 31 33 35 36