Question

rezolvare completa va rog

Answer 1

$\lim\limits_{x\to 0}\dfrac{e^x+e^{-x}-2\cos(ax)}{3x^2} = 2 \\ \\ \text{Aplicam L'H:} \\ \\ \lim\limits_{x\to 0}\dfrac{e^x-e^{-x}+2a\sin(ax)}{6x} = 2 \\ \\ \text{Aplicam L'H din nou:}\\\\\lim\limits_{x\to 0}\dfrac{e^x+e^{-x}+2a^2\cos(ax)}{6} = 2\\ \\ \Rightarrow \dfrac{1+1+2a^2}{6} = 2 \Rightarrow 2+2a^2 = 12 \Rightarrow 2a^2 = 10 \Rightarrow\\ \\ \Rightarrow a^2 = 5 \Rightarrow \boxed{a = \sqrt{5}}$