Algebra of Limits

Q. If $f\left(x\right)=\{\begin{array}{ccc}sinx,& x\ne n\pi ,& nϵI\\ 2,& otherwise& \end{array}$ and $g\left(x\right)=\{\begin{array}{ccc}{x}^2+1,& x\ne 0,& 2\\ 4,& x=0& \\ 5,& x=2& \end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$

1. 6
2. 7
3. 1
4. 5

Q.

Find the value of $\underset{x\to 2}{lim}\frac{x^2-4}{x+2}$

1. 4

2. 0

3. Limit does not exist

4. 2

Q.

Find the value of $\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{x}$

1. $0$

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

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

4. limit does not exist

Q.

$\underset{x\to \infty }{lim}\frac{{\cos }^{2}x}{x}=$

1. $0$

2. $1$

3. $-1$

4. $\infty$

Q.

$\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$
is valid if $\underset{x\to a}{lim}f\left(x\right)$ does not exists?

1. False

2. True

Q.

$\underset{x\to 1}{lim}f\left(x\right)=5,$ and $\underset{x\to 1}{lim}g\left(x\right)=2$ find the value of $\underset{x\to 1}{lim}\left(\right[g\left(x\right){]}^{f\left(x\right)})$ ___

Q.

If $\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=10$ and $\underset{x\to a}{lim}f\left(x\right)=2$, then find the value of $\underset{x\to a}{lim}g\left(x\right)$, provided that $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exists ___

Q.

If $\underset{x\to a}{lim}f\left(x\right)=l$ and $\underset{x\to a}{lim}g\left(x\right)=m$, then $\underset{x\to a}{lim}{\left(f\right(x\left)\right)}^{g\left(x\right)}=l^m$

1. True

2. False

Q.

$\underset{n\to \infty }{\overset{}{lim}}$ $\sum _{x=1}^{20}$ $cos{}^{2n}(x-10)$ is equal to

1. 0

2. 1

3. 19

4. 20

Q. Let $\alpha \left(a\right)$ and $\beta \left(a\right)$ be the roots of the equation $(\sqrt[3]{31+a}-1)+(\sqrt{1+a}-1)x+(\sqrt[6]{61+a}-1)=0$ where $a>-1$. then ${lim}_{a\to 0^{+}}\alpha \left(a\right)$ and ${lim}_{a\to 0^{+}}\beta \left(a\right)$ are

1. $-\frac{5}{2}\text{and}1$
2. $-\frac{1}{2}\text{and}-1$
3. $-\frac{7}{2}\text{and}2$
4. $-\frac{9}{2}\text{and}3$