Limits and Cont...

If $$\displaystyle \lim_{x\to0}{\displaystyle \frac{x^n - \sin^nx}{x - \sin^nx}}$$ is non-zero finite, then $$n$$ must be equal

$$\displaystyle \lim_{n \rightarrow \infty} \frac{-3n + (-1)^n}{4n - (-1)^n}$$ is equal to

$$\displaystyle \lim_{x\to1}{\displaystyle \frac{1-x^2}{\sin 2\pi x}}$$ is equal to

If $$\displaystyle \lim _{ x\to\infty  }\left\{\displaystyle \frac{x^3 + 1}{x^2 +1} - (ax + b)  \right\}  = 2$$, then

If $$\displaystyle f(x) = \left\{\begin{matrix}x^2+2, & x \geq 2\\ 1-x, & x < 2\end{matrix}\right.$$ and $$g(x) = \left\{\begin{matrix}2x, & x > 1\\ 3-x, & x \leq 1\end{matrix}\right.$$, then the value of $$\displaystyle \lim_{x \rightarrow 1} f(g(x))$$ is ............

$$\displaystyle \lim_{x\to2} \left( \left( \displaystyle \frac{x^3 - 4x}{x^3 - 8} \right)^{-1} - \left( \displaystyle \frac{x + \sqrt{2x}}{x - 2} - \displaystyle \frac{\sqrt {2}}{\sqrt{x} - \sqrt{2}} \right)^{-1} \right)$$ is equal to

$$\displaystyle f(x) = \frac{3x^2 + ax + a + 1}{x^2 + x - 2} $$ and $$\displaystyle \lim_{x \rightarrow - 2} f(x)$$ exists. 

$$\displaystyle \lim_{x\to0}\left( x^{-3}\sin{3x} + ax^{-2} + b \right)$$ exists and is equal to 0, then

$$\displaystyle \lim _{ x\to\infty } \left( \frac { x^{ 2 }+2x-1 }{ 2x^2-3x-2 }  \right) ^{\LARGE  \frac { 2x+1 }{ 2x-1 }  }$$ is equal to

$$\displaystyle \lim_{x \rightarrow 1} \frac{x^8 - 2x + 1}{x^4 - 2x +1}$$ equals