Limits and Deri...

$$\underset { x\rightarrow 0 }{ \lim } \dfrac { { 3 }^{ 2x }-{ 2 }^{ 3x } }{ x } $$ is equal to

Let $$f(x)=\dfrac{ax+b}{x+1},lim_{x\rightarrow 0} f(x)=2$$ and $$lim_{x\rightarrow \infty} f(x)=1$$ then $$f(-2)=$$

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

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 :

If  $$y ( x ):$$  Solution of a  $$D.E.$$

$$( x \log x ) \dfrac { d y } { d x } + y = 2 x \log x,$$   $$( x , 1 )$$
$$y ( e ) = ? \quad x = e$$

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?

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