Algebra of Limits

Q.

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0(a\ne 0)$ and $\alpha +h,\beta +h$ are the roots of $p{x}^2+qx+r=0(p\ne 0)$, then the ratio of the squares of their discriminants is

1. $a^2:p^2$

2. $a:p^2$

3. $a^2:p$

4. $a:2p$

Q. The solution set of $x^2-4x+1<0$ is

1. $[0,\infty )$
2. $\varphi \left(\text{empty set}\right)$
3. $R$
4. $(2-\sqrt{3},2+\sqrt{3})$

Q. If $f\left(x\right)=\frac{x+2}{x^2-8x-4}$, then which of the following is correct?

1. domain is $R-\{4-\sqrt{5},4+\sqrt{5}\}$
2. domain is $[4-\sqrt{5},4+\sqrt{5}]$
3. range is $(-\infty ,-\frac{1}{4}]\cup [-\frac{1}{20},\infty )$
4. range is $R$

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.

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

1. True

2. False

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 $L_1=\underset{x\to 2020}{lim}\sum _{k=1}^{2020}\frac{\{x+k\}}{2020}$ and $L_2=\underset{x\to 2020}{lim}(\left[x\right]+[-x]),$ then which of the following option is correct ?

(where $\left[x\right]$ and $\left\{x\right\}$ represent the greatest integer function and fractional part of $x$ respectively)

1. $L_1=0$ and $L_2$ does not exist
2. $L_1=0$ and $L_2=-1$
3. $L_1$ does not exist and $L_2=-1$
4. $L_1=2020$ and $L_2$ does not exist