Sandwich Theorem

Q. 19.How to find the value of sin (18) and cos (75). Angles are in degree.

Q.

For the function $f\left(x\right)=x\cos \left(\frac{1}{x}\right),x\ge 1$

1. For at least one $x$ in the interval $[1,\infty ],f(x+2)–f\left(x\right)<2$

2. $li{m}_{x\to \infty }f\left(x\right)=1$

3. For all $x$ in the interval $[1,\infty ],f(x+2)–f\left(x\right)>2$

4. $f\left(x\right)$ is strictly increasing in the interval $[1,\infty )$

5. $\mathrm{B},\mathrm{C}\&\mathrm{D}$

Q.

Evaluate $\underset{x\to 0}{lim}\left(\frac{3^{2x}-1}{2^{3x}-1}\right)$

Q. 30. Prove that , sin20.sin40.sin60.sin80 = 3/16 . Thanks.

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. $1^2+(1^2+2^2)+(1^2+2^2+3^2)+....$

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.

Evaluate:$\underset{x\to 0}{\lim }x{\log }_e\left(\sin x\right)$

1. $-1$

2. ${\log }_{e}1$

3. $1$

4. None of these

Q.

Evaluate $\underset{x\to 1}{lim}\frac{x^{15}-1}{x^{10}-1}$

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

1. 
2. 
3. does not exist
4. 

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

1. 
2. does not exist
3. 
4. 

Q. Write the value of $\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos 2x}}{x}.$

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. 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.

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

1. 0

2. -1

3. 1

4. 2

Q. lf $f\left(x\right)=\{\begin{array}{cc}-x-\frac{\pi }{2},& x\le -\frac{\pi }{2}\\ -cosx,& -\frac{\pi }{2}<x\le 0\\ x-1,& 0<x\le 1\\ lnx,& x>1\end{array}$ then

1. $f\left(x\right)$ ls contlnuous at $x=-\frac{\pi }{2}$
2. $f\left(x\right)$ is differentiable at $x=1$
3. $f\left(x\right)$ is differentiable at $x=-\frac{3}{2}$
4. $f\left(x\right)$ is not differentiable at $x=0$

Q. ${lim}_{x\to 0}\frac{(x+1{)}^5-1}{x}$

Q. Find the set of values of X satisfying 2 sin^2x-3sinx+1>=0

Q. Prove that $\left|\sin x\sin \left(\frac{\pi }{3}-x\right)\sin \left(\frac{\pi }{3}+x\right)\right|\le \frac{1}{4}$ for all values of x

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. 0

3. -1

4. does not exist

Q. $\frac{e^x+\sin x}{1+\log x}$