Given the function $f ( x ) = \left\{ \begin{array} { l l l } x e ^ { 2 x } & \text { if } & x < 0 \\ \frac { \ln ( x + 1 ) } { x + 1 } & \text { if } & x \geq 0 \end{array} \right.$, where $\ln$ means natural logarithm, it is requested:\ a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$.\ b) (1 point) Calculate $\lim _ { x \rightarrow - \infty } f ( x )$ and $\lim _ { x \rightarrow + \infty } f ( x )$.\ c) (1 point) Calculate $\int _ { - 1 } ^ { 0 } f ( x ) d x$
Given the function $f ( x ) = \left\{ \begin{array} { l l l } x e ^ { 2 x } & \text { if } & x < 0 \\ \frac { \ln ( x + 1 ) } { x + 1 } & \text { if } & x \geq 0 \end{array} \right.$, where $\ln$ means natural logarithm, it is requested:\
a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$.\
b) (1 point) Calculate $\lim _ { x \rightarrow - \infty } f ( x )$ and $\lim _ { x \rightarrow + \infty } f ( x )$.\
c) (1 point) Calculate $\int _ { - 1 } ^ { 0 } f ( x ) d x$