Definite Integral as Limit of Sum

Q.

$li{m}_{\pi \to \infty }{\sum }_{k=1}^n\frac{k}{n^2+k^2}$ is equals to [Roorkee 1999]

1. 
2. 
3. 
4. 

Q.

${lim}_{\pi \to \infty }\frac{1}{n}{\sum }_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}=$

1. $-1+\sqrt{5}$
   

2. $1+\sqrt{5}$

3. $-1+\sqrt{2}$
   

4. $1+\sqrt{2}$

Q. ${lim}_{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}{e}^{\frac{r}{n}}$ is

1. e
2. e - 1
3. 1 - e
4. 1 + e

Q.

${lim}_{\pi \to \infty }{\left[\frac{n!}{n^n}\right]}^{\frac{1}{n}}equals$ [Kurukshetra CEE 1998]

1. 
2. e

3. 1/e

4. 

Q.

The value of ${lim}_{n\to \infty }[\frac{n}{1+n^2}+\frac{n}{4+n^2}+\frac{n}{9+n^2}+\cdots +\frac{1}{2n}]$ is equal to [Bihar CEE 1994]

1. 1

2. 
   

3. 
   

4. None of these

Q. ${lim}_{n\to \infty }\frac{1}{1^3+n^3}+\frac{4}{2^3+n^3}+\cdots +\frac{1}{2n}$ is equal to

1. 
   

2. 
   

3. None of these

4. 
   

Q. For a first order reaction, the plot of against log C gives a straight line with a slope equal to:

1. $\left(\frac{k}{2}.303\right)$
2. $\left(\frac{-k}{2}.303\right)$
3. $\left(ln\frac{k}{2}.303\right)$
4. $-k$

Q.

$li{m}_{\pi \to \infty }\frac{1^{\rho }+2^{\rho }+3^{\rho }+\cdots +n^{\rho }}{n^{\rho +1}}=$ [AIEEE 2002]

1. 
   

2. 
   

3. 
   

4. 
   

Q.

$li{m}_{\pi \to \infty }{\sum }_{r=1}^n\frac{1}{n}{e}^{\frac{r}{n}}is$ [AIEEE 2004]

1. 
2. e
3. 
4. 

Q. $\stackrel{lim}{n\to \infty }[\frac{1}{n+1}+\frac{1}{n+2}+\cdots \frac{1}{n+n}]$ is equal to

1. 3 log 2
2. log 2
3. None of these
4. 2 log 2