\( 1 \leftarrow \) Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 33 in. by 18 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume. The dimensions of box of maximum volume are (Round to the nearest hundredth as needed. Use a comma to separate answers as needed.)

\( \int \frac { \int \ln ( 4 x ) } { 6 x ^ { 3 } } d x \)

Aufgabe 1 (3 Punkte). Bestimmen Sie die Grenzwerte der folgenden konvergenten Folgen \( \left(a_{n}\right) \) : \[ \begin{array}{l}\text { (a) } a_{n}=\frac{\sqrt[n]{7} n^{10} 2^{n}-3^{n+1} \sqrt[n]{n}}{3^{n}+2^{n} n^{10}} \quad\left(\text { b) } a_{n}=\sqrt{n^{2}+2 n}-n, \quad(c) a_{n}=\frac{b^{n}}{n!} \quad(b \in \mathbb{R})\right.\end{array} \]

Aufgabe \( 3(4 \) Punkte \( ) \). Die R-wertigen Folgen \( \left(a_{n}\right) \) und \( \left(b_{n}\right) \) seien gegeben durch \[ \left(a_{n}\right)=(0,1,2,1,0,1,2,1,0,1,2,1, \ldots), \quad b_{n}=(2,1,1,0,2,1,1,0,2,1,1,0, \ldots) \] Bestimmen Sie \( \quad \liminf _{n \rightarrow \infty} a_{n}+\liminf _{n \rightarrow \infty} b_{n}, \quad \liminf _{n \rightarrow \infty}\left(a_{n}+b_{n}\right), \quad \limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right), \quad \limsup _{n \rightarrow \infty} a_{n}+\limsup _{n \rightarrow \infty} b_{n} \)

Sea \( h(x)=\frac{1}{x^{3}}-\frac{2}{x^{2}} \) \( h^{\prime}(-1)=\square \)

احسب التكامل لـ \( \frac{4x^3 - 7x + 5}{2x + 1} dx \) من خلال تطبيق طريقة القسمة الطويلة.