Răspuns:
a = 2
b = 4
Explicație pas cu pas:
[tex]\displaystyle\lim\limits_{x\to 0} \frac{2e^{ax}-ae^{bx}}{x} = -4\\ \\\textrm{Avem la numitor 0 si valoarea limitei este finita}\implies \\\begin{cases}\lim\limits_{x\to 0}2e^{ax}-ae^{bx}=0\\ \lim\limits_{x\to 0}\frac{[2e^{ax}-ae^{bx}]'}{[x]'}=-4\end{cases}\\ \\ \lim\limits_{x\to 0}2e^{ax}-ae^{bx}=0\\ \\ a,b\in \mathbb{R}\implies 2-a = 0\implies \boxed{a = 2}\\ \\ \lim\limits_{x\to 0} 2e^{ax}\cdot a - ae^{bx}\cdot b = -4 \\ \\ a,b \in \mathbb{R}\implies \lim\limits_{x\to 0} 2e^{ax}\cdot a - ae^{bx}\cdot b = 2a - ab = -4\\ \\ \implies 2(2 - b) = -4\\ \\ 2-b=-2\\ \\\boxed{b=4}[/tex]
