The product of all possible values of , for which x 0lim(1- ( x) (( +1)x) (( +2)x) ^2(( +1…

Question

The product of all possible values of $\alpha $, for which

$\underset{x	o 0}{lim}(\frac{1-\cos (\alpha x)\cos ((\alpha +1)x)\cos ((\alpha +2)x)}{{\sin }^{2}((\alpha +1)x)})=2$, is :

Accepted Answer

$\underset{x	o 0}{lim}\frac{1-\cos \alpha x\cdot \cos ((\alpha +1)x)\cos ((\alpha +2)x)}{(\alpha +1{)}^{2}{x}^{2}}$

By using L.H. Rule we get

$\begin{gathered} \Rightarrow \frac{1}{2}(\alpha +1{)}^{2}[{\alpha }^{2}+(\alpha +1{)}^{2}+(\alpha +2{)}^{2}]=2 \ {\alpha }^{2}+(\alpha +1{)}^{2}+(\alpha +2{)}^{2}=4(\alpha +1{)}^{2} \ {\alpha }^{2}+(\alpha +2{)}^{2}=3(\alpha +1{)}^{2} \ \Rightarrow {\alpha }^{2}+2\alpha -1=0 \ =-1 \end{gathered}$

Solutions · 1