Definite Integral as Limit of Sum

Q.

${lim}_{\pi \to \infty }\frac{1^{99}+2^{99}+3^{99}+\cdots \cdots n^{99}}{n^{100}}=$ [EAMCET 1994]

1. $\frac{9}{100}$

2. $\frac{1}{100}$

3. $\frac{1}{99}$

4. $\frac{1}{101}$

Q. Show that $\underset{x\to 0}{\lim }\frac{1}{x}$ does not exist.

Q.

Evaluate $\underset{x\to 2}{lim}\left(\frac{e^{3x}-1}{x}\right)$

Q. $\underset{n\to \infty }{lim}(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+...+\frac{1}{5n})$

is equal to :

1. ${\tan }^{-1}\left(2\right)$
2. ${\tan }^{-1}\left(3\right)$
3. $\pi /4$
4. $\pi /2$

Q.

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

1. $\frac{1}{p+1}$

2. $\frac{1}{p-1}$

3. $\frac{1}{p}-\frac{1}{p-1}$

4. $\frac{1}{p+2}$

Q. $\text{Let}f\left(x\right)=\underset{n\to \infty }{lim}{\left(\frac{n^n(x+n)(x+\frac{n}{2})\cdots (x+\frac{n}{n})}{n!(x^2+n^2)(x^2+\frac{n^2}{4})\cdots (x^2+\frac{n^2}{n^2})}\right)}^{\frac{x}{n}},$ for all $x>0.$ Then

1. $f\left(\frac{1}{2}\right)\ge f\left(1\right)$
2. $f\left(\frac{1}{3}\right)\le f\left(\frac{2}{3}\right)$
3. $f^{\prime }\left(2\right)\le 0$
4. $\frac{f^{\prime }\left(3\right)}{f\left(3\right)}\ge \frac{f^{\prime }\left(2\right)}{f\left(2\right)}$

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. $\frac{\pi }{2}$

2. $\frac{\pi }{4}$

3. $1$

4. $Noneofthese$

Q.

${lim}_{n\to \infty }[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{2n}]=$ [Karnataka CET 1999]

1. $0$
2. $lo{g}_{e}4$

3. $lo{g}_{e}3$

4. $lo{g}_{e}2$

Q. The value of $\int \frac{x^2-1}{(x^2+1)\sqrt{1+x^4}}dx$ is (where $C$ is integration constant)

1. ${\sec }^{-1}\left(\frac{\sqrt{x^2+\frac{1}{x^2}}}{\sqrt{2}}\right)+C$
2. $\frac{1}{2\sqrt{2}}{\sec }^{-1}\left(\frac{\sqrt{x^2+\frac{1}{x^2}}}{\sqrt{2}}\right)+C$
3. $\frac{1}{\sqrt{2}}{\sec }^{-1}\left(\frac{x+\frac{1}{x}}{\sqrt{2}}\right)+C$
4. $\frac{1}{2}{\sec }^{-1}\left(\frac{\sqrt{x^2+\frac{1}{x^2}}}{\sqrt{2}}\right)+C$

Q. If usinx + vcosx=5 anf ucosx-vsinx=7 then u"v - uv" equals (the differential is with respect to x) A) - 74(v)³/u³ B) 74 v (v)²/u³ C) 1 D) 74u (v)²

Q. $\underset{n\to \infty }{lim}{\sum }_{r=1}^n\frac{n}{n^2+r^2{x}^2},x>0$is equal to :

1. $ta{n}^{-1}x$
2. $xta{n}^{-1}x$
3. $\frac{ta{n}^{-1}x}{x}$
4. $\frac{ta{n}^{-1}x}{x^2}$

Q.

$li{m}_{\pi \to \infty }\frac{1}{n}{\sum }_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}equals$ [IIT 1997 Re-exam]

1. $1+\sqrt{5}$
2. $-1+\sqrt{5}$
3. $-1+\sqrt{2}$
4. $1+\sqrt{2}$

Q. The value of $\underset{n\to \infty }{lim}[\frac{n}{(n+1)(n+2)}+\frac{n}{(n+2)(n+4)}+\cdots +\frac{1}{6n}]$ is:

1. $\ln 2$
2. $\ln \frac{3}{5}$
3. $\ln 3$
4. $\ln \frac{3}{2}$