$\lim _{x \rightarrow 0} \frac{\left(1+\frac{x}{2}\right)^{5 / 7}-1}{x} = $

The value of the limit of $\frac{x^3 - x^2 - 18}{x - 3}$ as $x$ tends to $3$ is

$\lim _{x \rightarrow 0} \frac{\cos 2x - \cos 3x}{\cos 4x - \cos 5x} = $

$\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{{e^x} - {e^{\sin x}}}}{{x - \sin x}}} \right]$ is equal to

$A$ 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:

$\lim _{x \rightarrow 0} \left( \frac{3^{x}-1}{x} \right)$ is equal to