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

Given the system of differential equations: \[ \frac{d \mathbf{x}}{d t}=A \mathbf{x} \] where \( A=\left(\begin{array}{ll}4 & 0 \\ 0 & 3\end{array}\right) \), what is the general solution for \( \mathbf{x}(t) \) ?

Analizar en cada caso si es posible aplicar el Teorema de Stokes para calcular la circulación de \( \vec{F} \) a lo largo de \( C:\left\{\begin{array}{l}x^{2}+y^{2}=1 \\ z=0\end{array}\right. \) Hallar el valor de esa circulación. (i) \( \vec{F}=\left\langle\frac{-x}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}, \frac{-y}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}, \frac{-z}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}\right\rangle \)

Encontrar \( f(t) \) si \( F(s)=\frac{2}{(s+2)^{4}}+\frac{3}{s^{2}+16}+\frac{5(s+1)}{s^{2}+2 s+5} \)

Determine la longitud de la curva \( x=\frac{y^{5}}{5}+\frac{1}{12 y^{3}} \) en el intervalo \( y \in[1,2] \). resultado con dos decimales.

One measure of living standards in country \( A \) is given by the following formula \( L(t)=7+4 e^{0.14 t} \) In this formula \( t \) is the number of years since 1982. Find \( L \) to the nearest hund \( \begin{array}{ll}\text { (a) } 1988 & \text { (b)1998 } \\ \text { (c)2007 }\end{array} \) \( \begin{array}{l}\text { (a) Find } L \text { for } 1988 . \\ \text { (Round to the nearest hundredth as needed.) } \\ \text { (b) Find } L \text { for } 1998 . \\ L=\square\end{array} \) (Round to the nearest hundredth as needed.) (c) Find \( L \) for 2007 .