Limits and Deri...

The value of $$lim_{x \to 0} (\dfrac{1}{x^2} - cotx)$$ equals 

$$ \displaystyle \lim _{ x-\infty  }{ sgn\left( \cot{\dfrac { { \pi x }^{ 2019 } }{ { x }^{ 2019 }+7 }}  \right)  }$$

$$\displaystyle\lim_{x \to \pi/2} (sec x +tan x)$$ is equal to 

$$\underset { x\rightarrow 0 }{ lim } \dfrac { x\tan { 2x } -2\tan { 2x }  }{ { \left( 1-cos2x \right)  } }$$ equals:

$$\underset{x \rightarrow 0}{Lt} \dfrac{sin x - x + x^3 / 6}{x^5}$$ = 

$$\displaystyle \lim_{x\rightarrow0 }{\dfrac{(\cos\alpha)^{x}-(\sin\alpha)^{x}-\cos 2\alpha}{(x-4)}}, \alpha\in \left(0, \dfrac{\pi}{2}\right)$$ is equal to

$$\displaystyle\lim_{x\to \pi/2} \dfrac{sinx-(sinx)^{sin x}}{1-sin x + In sin x}$$ is equal to

$$lim_{n\to \infty} \Sigma^n_{r=1} \dfrac{\pi}{n} sin(\dfrac{\pi r}{n})$$ is equal to

If $$\mathop {\lim }\limits_{x \to 0} \frac{{x\left( {1 + a\cos x} \right) - b\sin x}}{{{x^3}}} = 1,$$ then

Evaluate : $$\displaystyle\lim _{ x\rightarrow 0 }{ \left( \dfrac { { e }^{ x\ell n\left( { 3 }^{ x }-1 \right)  }-\left( { 3 }^{ x }-1 \right) ^{ x }\sin { x }  }{ { e }^{ x\ell nx } }  \right)  } $$ is equal to