Limit

Q.

Let $f:R\to R:f\left(x\right)=x^2$ and $g:R\to R:g\left(x\right)=2x+1$. Find

(i) $(f+g)\left(x\right)$

(ii) $(f-g)\left(x\right)$

(iii) $\left(fg\right)\left(x\right)$

(iv) $\left(\frac{f}{g}\right)\left(x\right)$

Q. Which of the following pairs of sets are disjoint
(i) {$1,2,3,4$} and {$x:x$ is a natural number and $4\le x\le 6$}
(ii) { $a,e,i,o,u$ } and { $c,d,e,f$ }
(iii) {$x:x$ is an even integer } and {$x:x$ is an odd integer}

Q. $li{m}_{x\to \frac{\pi }{4}}\frac{1-co{t}^{3}x}{2-cotx-co{t}^{3}x}=$___

1. 0
2. $\frac{3}{4}$
3. $\infty$
4. $\frac{1}{2}$

Q.

Show that $\underset{x\to 0}{lim}\frac{x}{|x|}$ does not exist.

Q. If $a_r=(\cos 2r\pi +i\sin 2r\pi {)}^{\frac{1}{9}}$, then the value of $\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ is:

1. $-1$
2. $1$
3. $0$
4. $2$

Q.

${lim}_{x\to 1}(\frac{2}{1-x^2}+\frac{1}{x-1})=$___

1. $\frac{1}{2}$

2. 1

3. $\infty$

4. 0

Q.

Work out the following divisions and find out Quotient and Remainder:

$2\overline{)2465}$

Q. If $f\left(x\right)=|x|-5$ , evaluate the following limits: