$\lim _{\theta \rightarrow \frac{\pi}{2}^{-}} \frac{8 \tan ^4 \theta+4 \tan ^2 \theta+5}{(3-2 \tan \theta)^4} = $

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x^2}-\sqrt{1-x+x^2}}{3^x-1}$ is equal to

The value of $\mathop {\lim }\limits_{x \to \infty } \frac{{{x^2}\sin \frac{1}{x} - x}}{{1 - |x|}}$ is

$\mathop {\lim }\limits_{x \to {a^ + }} \left( \frac{{|x{|^3}}}{a} - {\left[ {\frac{x}{a}} \right]^3} \right) \,(a > 0)$ is equal to :- (where $[x]$ is the greatest integer function and $|x|$ is the modulus function)

If $a$ is the minimum value of $\sin^2 \theta - \sin \theta + \frac{1}{2}$ and $b = \lim_{x \to \infty} (\sqrt{(x + 1)(x + 2)} - x)$,then $|2a + b| = $

If $f(x) = \frac{x(a^x - 1)}{1 - \cos x}$ and $g(x) = \frac{x(1 - a^x)}{a^x(\sqrt{1 - x^2} - \sqrt{1 + x^2})}$,then $\lim_{x \to 0} (f(x) - g(x)) = $