Limits and Cont...

Evaluate the limit, $$\mathop {\lim }\limits_{x \to 0} \frac{{x({{(1 + x)}^{1/x}} - e)}}{{x({{(1 + {x^2})}^{1/{x^2}}} - e)}}$$

Let  $$U_{ { n } }=\dfrac { n! }{ (n+2)! } $$  where  $$n \in N .$$  If  $$S_{ { n } }=\sum _{ { n-1 } }^{ { n } } U_{ { n } }$$  then  $$\lim _ { n \rightarrow \infty } \mathrm { S } _ { n }$$  equals :

$$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } - x + 1 } - a x - b \right) = 0,$$   then the values of  $$a$$  and  $$b$$  are given by

The value of $$\lim_{x\rightarrow \infty }$$ y In $$(\frac{sin (x+1/y)}{sin x})$$ when $$0 < x < \pi /2$$ is

$$\displaystyle\lim_{n\rightarrow\infty}\left\{\dfrac{n!}{(kn)^n}\right\}^{\dfrac{1}{n}}, k\neq 0$$, is equal to?

Let $$f(x)=\displaystyle\lim_{n\rightarrow \infty}\sum^{n-1}_{r=0}\dfrac{x}{(rx+1)\{(r+1)x+1\}}$$, then?

$$\displaystyle \lim _{ x\rightarrow \infty }{ \left[\dfrac{n}{n^{2}+1^{2}}+\dfrac{n}{n^{2}+2^{2}}+\dfrac{n}{n^{2}+3^{2}}+....+\dfrac{1}{n^{5}}\right] }$$

$$\displaystyle\lim _{ x\rightarrow 0 }{ { x }^{ 2 }{ e }^{ \sin { \frac { 1 }{ x }  }  } } $$ equals 

Evaluate: $$\underset { { x\rightarrow}\dfrac{ \pi  }{ 4 }  }{ lim } \dfrac { { cot }^{ 3 }x-tanx }{ cos\left( x+\pi /4 \right)  } \quad $$