1) Given the vector-valued function r(t)=⟨e−t,−et−t,tln⁡t⟩ a- Evaluate limt⇒0r(t) and Evaluate limt⇒∞r(t) b- Find r...

a- Evaluate the limits
${\displaystyle \underset{t\Rightarrow 0}{lim}r\left(t\right)=\underset{t\Rightarrow 0}{lim}⟨e^{-t},-t{e}^{-t},t\ln t⟩}$
${\displaystyle =⟨\underset{t\Rightarrow 0}{lim}{e}^{-t},\underset{t\Rightarrow 0}{lim}-t{e}^{-t},\underset{t\Rightarrow 0}{lim}t\ln t⟩}$
${\displaystyle =⟨e^0,\left(0\right)e^0,\left(0\right)\ln 0⟩}$
${\displaystyle =⟨1,0,0⟩}$
${\displaystyle \underset{t\Rightarrow \mathrm{\infty }}{lim}r\left(t\right)=\underset{t\Rightarrow \mathrm{\infty }}{lim}⟨e^{-t},-t{e}^{-t},t\ln t⟩}$
${\displaystyle =⟨\underset{t\Rightarrow \mathrm{\infty }}{lim}{e}^{-t},\underset{t\Rightarrow \mathrm{\infty }}{lim}-t{e}^{-t},\underset{t\Rightarrow \mathrm{\infty }}{lim}t\ln t⟩}$
$=\mathrm{\infty }$
The limit does not exist because as tapproaches infinity, the limit of tIn(t) does not exist.
b- Find r'(t) and evaluate r'(0)
${\displaystyle r\left(t\right)=\frac{d}{dt}⟨e^{-t},-t{e}^{-t},t\ln t⟩}$
${\displaystyle =⟨\frac{d}{dt}\left(e^{-t}\right),\frac{d}{dt}(-t{e}^{-t}),\frac{d}{dt}\left(t\ln t\right)⟩}$
${\displaystyle =⟨e^{-t},=e^{-t}+t{e}^{-t},\ln \left(t\right)+1⟩}$
${\displaystyle r^{\prime }\left(t\right)=⟨-e^{-0},-e^0+{\left(0\right)}^{-0},\ln \left(0\right)+1⟩}$
${\displaystyle =⟨-1,-1,\mathrm{\infty }⟩}$
The value of r’(0) is undefined
c- Find r''(t)
We have ${\displaystyle r^{\prime }\left(t\right)=⟨-e^{-t},e^{-t}+t{e}^{-t},\ln \left(t\right)+1⟩}$
${\displaystyle rt)=\frac{d}{dt}⟨-e^{-t},e^{-t}+t{e}^{-t}}$