(21) \( f(x)=\left\{\begin{array}{ll}x^{2} & x \leq-1 \\ 2 x+3 & -1<x<0 \\ 4 & x \geq 0\end{array}\right. \)

\( F ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \log _ { a } ( x + h ) - \log _ { a } ( x ) } { h } \)

Una sucesión puede expresarse como un límite. De esta manera \( \frac{i n(n)}{n} \) es: \( \begin{array}{l}\text { a. }(\ln ) / n=-1 \\ \left.\text { b. }(\ln )^{n}\right) / n=1 \\ \text { c. }(\ln ,) / n=0 \\ \text { d. }(\ln ), n=\infty\end{array} \)

\( \lim ( 2 + \frac { 2 } { n } ) ^ { m } \)

\[ f(x)=\frac{x^{2}+3 x+2}{x^{2}-2 x-3} \] a) Find the domain of \( f \) and deduce the vertical asymptote. b) Find \( x \)-axis intercepts, the \( y \)-axis intercepts and the horizontal asymptote ( check the intersection with the horizontal asymptote). c) Compute the derivative \( f^{\prime} \) of \( f \) and analyze it d) Compute the second derivative \( f^{\prime \prime} \) of \( f \) and analyze it. e) sketch the graph of \( f \)

b) \( F^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\log _{2}(x+h)-\log _{2}(x)}{h} \)