Question

Solve:
$\underset{x\to 0}{lim}\left(\cos ecx-\cos x\right)$

Answer 1

$\underset{x\to 0}{lim}(\csc x-\cos x)$

$\Rightarrow \underset{x\to 0}{lim}(\frac{1}{\sin x}-\cos x)$

$\Rightarrow \underset{x\to 0}{lim}\left(\frac{1-\sin x\cos x}{\sin x}\right)$

this is of the form $\%$ we use L' Hospital rule

$\Rightarrow \underset{x\to 0}{lim}\left(\frac{-{\cos }^{2}x+{\sin }^{2}x}{\cos x}\right)$

$\Rightarrow \underset{x\to 0}{lim}-\frac{(\cos 2x}{\left(\cos x\right)}$

$\Rightarrow \frac{-\left(\cos 0\right)}{\left(\cos 0\right)}\Rightarrow -1$

$\underset{x\to 0}{lim}(\csc x-\cos x)=-1$