Question

If $\underset{x\to a}{lim}\left(f\right(x)+g(x\left)\right)=2$ and $\underset{x\to a}{lim}\left(f\right(x)-g(x\left)\right)=1$,

then the value of $\underset{x\to a}{lim}f\left(x\right)g\left(x\right)$ is?

• A. Does not exist
• B. Exists and is $\frac{3}{4}$
• C. Exists and is $-\frac{3}{4}$
• D. Exists and is $\frac{4}{3}$

Answer 1

Answer: (B)

Exists and is $\frac{3}{4}$

Given, $\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=2$
$\Rightarrow {lim}_{x\to a}f\left(x\right)+{lim}_{x\to a}g\left(x\right)=2$ .....(i)
$\underset{x\to a}{lim}\left[f\right(x)-g(x\left)\right]=1$
$\Rightarrow {lim}_{x\to a}f\left(x\right)-{lim}_{x\to a}g\left(x\right)=1$ ......(ii)

Adding (i) and (ii), we get
${lim}_{x\to a}f\left(x\right)=\frac{3}{2}$
${lim}_{x\to a}g\left(x\right)=\frac{1}{2}$
${lim}_{x\to a}f\left(x\right)g\left(x\right)={lim}_{x\to a}f\left(x\right).{lim}_{x\to a}g\left(x\right)=\frac{3}{4}$