Let f and g be real valued functions defined on interval $(-1,1)$ such that $g\text{'}\text{'}\left(x\right)$ is continuous, $g\left(0\right)\ne 0,g\text{'}\left(0\right)=0,g\text{'}\text{'}\left(0\right)\ne 0$ and $f\left(x\right)=g\left(x\right)\sin x$ then $\underset{x\to 0}{\lim }(g\left(x\right)\cot x-g\left(0\right)\cos ecx)=$   

$f\text{'}\left(x\right)=g\text{'}\left(x\right)\sin x+g\left(x\right)\cos x$ 
$f\text{'}\left(0\right)=g\left(0\right)$ 
$f\text{'}\text{'}\left(x\right)=g\text{'}\text{'}\left(x\right)\sin x+g\text{'}\left(x\right)\cos x+g\text{'}\left(x\right)\cos x-g\left(x\right)\sin x$ 
$f\text{'}\text{'}\left(0\right)=2g\text{'}\left(0\right)=0$ 
$\underset{x\to 0}{\lim }g\left(x\right)\cot x-g\left(0\right)\cos ecx=\underset{x\to 0}{\lim }\frac{g\left(x\right)\cos x-g\left(0\right)}{\sin x}$ 
$=\underset{x\to 0}{\lim }\frac{g\text{'}\left(x\right)\cos x+g\left(x\right)(-\sin x)}{\left(\cos x\right)}$ 
$=g\text{'}\left(x\right)=0=f\text{'}\text{'}\left(0\right)$