Limits and Cont...

$$\mathop {\lim }\limits_{x \to 0} \dfrac{{{e^{4x}} - 1}}{x}$$

$$\displaystyle \lim_{n \rightarrow \infty}\dfrac {1^{2}+2^{2}+3^{2}+....+n^{2}}{n^{3}}$$ is equal to

$${ x }_{ n }={ \left( 1-\cfrac { 1 }{ 3 }  \right)  }^{ 2 }{ \left( 1-\cfrac { 1 }{ 6 }  \right)  }^{ 2 }{ \left( 1-\cfrac { 1 }{ 10 }  \right)  }^{ 2 }...{ \left( 1-\cfrac { 1 }{ \cfrac { n(n+1) }{ 2 }  }  \right)  }^{ 2 },n\ge 2$$. Then the value of $$\displaystyle\lim _{ n\rightarrow \infty  }{ { x }_{ n } } $$

$$ \lim _{ x\rightarrow 1 }{ \dfrac { \sqrt { 1-\cos { 2\left( x-1 \right)  }  }  }{ x-1 }  }$$

The value of k which makes $$f(x)=\left\{\begin{matrix} \sin\dfrac{1}{x}, x\neq 0\\ k, x=0\end{matrix}\right.$$ continuous at $$x=0$$ is?

If $$f(x) = 3x^{10} - 7x^8 + 5x^6 - 21x^3 + 3x^2 - 7$$, then $$\underset{a \rightarrow 0}{\lim} \dfrac{f(1 - \alpha) - f(1)}{\alpha^3 + 3 \alpha}$$ is 

Let $$f(x) = \underset{0}{\overset{x}{\int}} |2t - 3| dt$$, then $$f$$ is 

If $$f(x) = 2x^9 - 5x^8 + 7x^6 - 15x^4 + 5x + 7$$, then $$\underset{x \rightarrow 0}{\lim} \dfrac{f (1 - \alpha) - f(1)}{\alpha^3 + 3 \alpha}$$ is 

The value of $$\lim_{x\rightarrow o}\dfrac{\sqrt{\dfrac{1}{2}(1-cos 2 x)}}{x}$$

The value of $$\underset { x\longrightarrow \infty  }{ Lim } \dfrac{d}{dx}\overset { \sqrt { 3 }  }{ \underset { -\sqrt { 3 }  }{ \int }  } \dfrac{r^3}{(r+1)(r-1)}$$dr,is