If `lim_(xtoa)[f(x)+g(x)]=2` and `lim_(xtoa) [f(x)-g(x)]=1,` then find the value of `lim_(xtoa) f(x)g(x).`

Explanation

`underset(xtoa)lim[f(x)+g(X)]=2`
or ` underset(xtoa)limf(x)+underset(xtoa)limg(x)=2" "(1)`
`underset(xtoa)lim[f(x)-g(x)]=1`
or `underset(xtoa)limf(x)-underset(xtoa)limg(x)=1" "(2)`
Adding (1) and (2),
`2underset(xtoa)limf(x)=3" "or" "underset(xtoa)limf(x)=3/2`
Subtracting (2) from (1),
`2underset(xtoa)limg(x)=1" "`or`" "underset(xtoa)limg(x)=1/2`
or `" "underset(xtoa)limf(x)g(x)=underset(xtoa)limf(x)underset(xtoa)limg(x)=3/2xx1/2=3/4`