Limits and Cont...

Consider $$f(x)=\lim _{ n\rightarrow \infty  }{ \cfrac { { x }^{ n }-\sin { { x }^{ n } }  }{ { x }^{ n }+\sin { { x }^{ n } }  }  } $$ for $$x>0,x\neq 1,f(1)=0$$ then

If $$\sum _{ r=1 }^{ k }{ \cos ^{ -1 }{ \beta  }  } =\cfrac { k\pi  }{ 2 } $$ for any $$k\ge 1$$ and $$A=\sum _{ r=1 }^{ k }{ { \left( { \beta  }_{ r } \right)  }^{ r } } $$, then $$\lim _{ x\leftarrow A }{ \cfrac { { \left( 1+x \right)  }^{ 1/3 }-{ \left( 1-2x \right)  }^{ 1/4 } }{ x+{ x }^{ 2 } }  } $$ is equal to

$$\displaystyle \lim_{x \rightarrow 0} \frac{ae^x + b cos x + c. e^{-x}}{sin^2 x} = 4$$ then b =

$$\lim_{x\rightarrow \dfrac{\pi}{6}} \dfrac{2 sin^{2} x + sin x - 1}{2 sin^{2} x - 3 sin x + 1}$$

$$\mathop {\lim }\limits_{x \to {a^ + }} {{\left\{ x \right\}\sin \left( {x - a} \right)} \over {{{\left( {x - a} \right)}^2}}}$$ 

is equal to (where {.} denotes the fraction
part of x and $$a \in N$$

Let$$f(\theta) = \dfrac{1}{tan^{9}\theta} {(1+tan\theta)^{10}+(2+tan\theta)^{10}+....+(20+tan\theta)^{10}}-20tan\theta$$. The left hand limit of $$f(\theta)$$ as $$\theta \rightarrow \dfrac{\pi}{2}$$ is:

If $$l=\lim\limits_{n\to 3}\dfrac{x^2-9}{\sqrt{x^2+7}-4}$$ and $$m=\lim\limits_{n\to -3}\dfrac{x^2-9}{\sqrt{x^2+7}-4}$$, then

If $$f : R \rightarrow R$$ is defined by
$$f(x) = \left \{\begin{matrix} \dfrac{x + 2}{x^2 + 3x + 2} & if & x \in R - \{-1, -1\} \\ -1 & if & x = -2 \\ 0 & if  &x = -1\end{matrix} \right.$$ then $$f(x)$$ continuous on the set 

If $$f\left( x \right) = \left\{ \begin{array}{l}\frac{{1 - \left| x \right|}}{{1 + x}},{\rm{ }}x \ne  - 1\\1,{\rm{          }}x =  - 1{\rm{     }}\end{array} \right.$$   then $$f\left( {\left[ {2x} \right]} \right),$$ where $$\left[ {} \right]$$ represents the greatest integer function , is 

if $$f\left( x \right) = \left\{ {\matrix{   {\cos \left[ x \right],} & {x \ge 0}  \cr    {\left| x \right| + a,} & {x < 0}  \cr 
 } } \right\}$$ Find
the value of a , given that $$\mathop {\lim }\limits_{x \to 0} f\left( x \right)$$  exists,
where[.]  denotes