Question

Given $\underset{x\to a}{lim}f\left(x\right)=2$ and $\underset{x\to a}{lim}g\left(x\right)=3$

Match th columns

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ 1.{lim}_{x\to a}\left(f\right(x)\times g(x\left)\right)& \text{P.8}\\ 2.{lim}_{x\to a}\left(f\right(x)-g(x\left)\right)& Q.10\\ 3.{lim}_{x\to a}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)\times 15& R.-1\\ 4.{lim}_{x\to a}\left(f\right(x{)}^{g\left(x\right)})& S.6\end{array}$

• A. 1 - P, 2 - Q, 3 - S, 4 - R
• B. 1 - Q, 2 - P, 3 - R, 4 - S
• C. 1 - Q, 2 - P, 3 - S, 4 - R
• D. 1 - S, 2 - R, 3 - Q, 4 - P
• E. 1 - S, 2 - R, 3 - P, 4 - Q

Answer 1

The correct option is D

1 - S, 2 - R, 3 - Q, 4 - P

If ${lim}_{x\to a}f\left(x\right)=L$ and ${lim}_{x\to a}g\left(x\right)=m$ then ${lim}_{x\to a}\left(f\right(x{)}^{g\left(x\right)})=L^m$. This is true for other operations like

multiplication, division, subtraction and addition. We can apply the limit independently.

$\Rightarrow 1\left){lim}_{x\to a}\right(f\left(x\right)\times g\left(x\right))={lim}_{x\to a}f(x)\times {lim}_{x\to a}g(x)$

$=1\times 3=6$

$2\left){lim}_{x\to a}\right(f\left(x\right)-g\left(x\right))=2-3$

$=-1$

$3){lim}_{x\to a}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)\times 15=15\times \frac{2}{3}$

$4\left){lim}_{x\to a}\right(f\left(x{)}^{g\left(x\right)}\right)=2^3=8$