The product of all possible values of , for which [_x0((1-( x)((+1)x)((+2)x))/(^2((+1)x)))…

Question

The product of all possible values of $\alpha$, for which
$$\lim_{x\to0}\left(\frac{1-\cos(\alpha x)\cos((\alpha+1)x)\cos((\alpha+2)x)}{\sin^2((\alpha+1)x)}\right)=2$$
is:

Accepted Answer

Using the standard small-angle result $1-\cos A\cos B\cos C\approx\dfrac{A^2+B^2+C^2}{2}$ (for $A,B,C\to0$), with $A=\alpha x,B=(\alpha+1)x,C=(\alpha+2)x$, and $\sin^2((\alpha+1)x)\approx(\alpha+1)^2x^2$:

$$\lim_{x\to0}\frac{\frac12[\alpha^2+(\alpha+1)^2+(\alpha+2)^2]x^2}{(\alpha+1)^2x^2}=2$$

$$\alpha^2+(\alpha+1)^2+(\alpha+2)^2=4(\alpha+1)^2 \implies \alpha^2+(\alpha+2)^2=3(\alpha+1)^2$$

Expanding: $2\alpha^2+4\alpha+4=3\alpha^2+6\alpha+3\implies \alpha^2+2\alpha-1=0$

By Vieta's formulas, the product of the roots is $\dfrac{-1}{1}=-1$.

Solutions · 1