$\mathop {\lim }\limits_{\alpha \to \beta } \left[ {\frac{{{{\sin }^2}\alpha - {{\sin }^2}\beta }}{{{\alpha ^2} - {\beta ^2}}}} \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:

$\lim _{x \rightarrow 0} \frac{\tan x - \sin x}{x^3}$ is equal to

$\mathop {\lim }\limits_{x \to 0} \frac{{{e^{\tan x}} - {e^x}}}{{\tan x - x}} = $

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^x-e^{\sin x}}{2(x-\sin x)}$

$\mathop {\lim }\limits_{x \to \pi /2} \left[ {x\tan x - \left( {\frac{\pi }{2}} \right)\sec x} \right] = $