Sandwich Theorem

Q. If $I=\sum _{k=1}^{98}\underset{k}{\overset{k+1}{\int }}\frac{k+1}{x(x+1)}dx,$ then

1. $I>{\log }_{e}99$
2. $I<{\log }_{e}99$
3. $I<\frac{49}{50}$
4. $I>\frac{49}{50}$

Q.

The value of $\underset{x\to 0}{lim}\left(\right[\frac{100x}{sinx}]+[\frac{99sinx}{x}\left]\right)$, where [.] denotes the greatest integer function, is

1. 197

2. 198

3. 199

4. Does not exist

Q. $\underset{x\to \infty }{lim}\sqrt{\frac{x+\sin x}{x-\cos x}}=$

1. $0$
2. $1$
3. $-1$
4. does not exist

Q.

$\text{The value of}\underset{x\to \infty }{lim}\frac{x+cosx}{x+sinx}is$

1. -1

2. 0

3. 1

4. 2