[tex]\\ Punctul \:\ a:
\\ \boxed{\left(f\cdot g\right)'=f'\cdot g+f\cdot g' =\ \textgreater \ f=(x-2),\:g=e^x}
\\ =\ \textgreater \ (x-2)' \cdot e^x + (x-2) \cdot e^x'
\\ =\ \textgreater \ (1-0) \cdot e^x + (x-2) \cdot e^x
\\ =\ \textgreater \ e^x +(x-2) \cdot e^x
\\ =\ \textgreater \ e^x(1+x-2)
\\ =\ \textgreater \ e^x (x-1)
\\ =\ \textgreater \ \boxed{f'_{(x)} = (x-1)e^x} [/tex]
[tex]\\ Punctul \:\ b:
\\ \lim _{x\to -\infty }\left(x-2\right)e^x =\ \textgreater \ \lim _{x\to -\infty } \frac{x-2}{e^{-x}} =\ \textgreater \ \frac{-\:\infty }{e^{-\left(-\infty \right)}} =\ \textgreater \ \frac{-\infty}{- (-\infty)}
\\ =\ \textgreater \ L`Hopital = \frac{(x-2)'}{ e^{-x}'} = \frac {1}{-e^{-x}} =\frac{1}{-e^{-\left(-\infty \right)}} = 0 [/tex]
