$\lim _{x \rightarrow 0} \frac{\log _{e}(1+x)}{3^{x}-1}$ is equal to

$\mathop {\lim }\limits_{x \to 0} \frac{{\sin x - x}}{{{x^3}}} = $

$\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x}}{{{x^3}}}$ is equal to

$\lim _{x \rightarrow 0} \left( \frac{x \cdot 10^x - x}{1 - \cos x} \right)$ is equal to

$\lim _{x \rightarrow 0} \left[ \frac{1}{x} - \frac{1}{e^x - 1} \right] = $

$A$ weight hangs by a spring and is caused to vibrate by a sinusoidal force. Its displacement $s(t)$ at time $t$ is given by an equation of the form $s(t) = \frac{A}{c^2 - k^2} (\sin kt - \sin ct)$,where $A, c,$ and $k$ are positive constants with $c \neq k$. Then the limiting value of the displacement as $c \to k$ is: