Sandwich Theorem

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

1. 
2. does not exist
3. 
4. 

Q. For $\alpha ϵR$ (the set of all real numbers), $a\ne -1$,
${lim}_{n\to \infty }\frac{(1^a+2^a+\dots +n^a)}{(n+1{)}^{a-1}[(na+1)+(na+2)+\dots +(na+n)]}=\frac{1}{60}$

1. $5$
2. $7$
3. $\frac{-15}{2}$
4. $\frac{-17}{2}$

Q.

If

$f\left(x\right)=\{\begin{array}{cc}xsin\frac{1}{x}& x\ne 0\\ 0& x=0\end{array},$

then $\underset{x\to 0}{lim}f\left(x\right)=$

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. 0

2. -1

3. 1

4. 2

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. 199

2. 197

3. 198

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

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. 198

2. 197

3. 199

4. Does not exist

Q.

If

$f\left(x\right)=\{\begin{array}{cc}xsin\frac{1}{x}& x\ne 0\\ 0& x=0\end{array},$

then $\underset{x\to 0}{lim}f\left(x\right)=$

1. -1

2. 1

3. 0

4. does not exist

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

1. 
2. 
3. 
4. does not exist

Q. For $\alpha ϵR$ (the set of all real numbers), $a\ne -1$,
${lim}_{n\to \infty }\frac{(1^a+2^a+\dots +n^a)}{(n+1{)}^{a-1}[(na+1)+(na+2)+\dots +(na+n)]}=\frac{1}{60}$

1. $5$
2. $7$
3. $\frac{-15}{2}$
4. $\frac{-17}{2}$