f(x) and g(x) are continuous functions such that limx→a[3f(x)+g(x)]=6 and limx→a[2f(x)−g(x)]=4. Given that the function h(x).g(x) is continuous at x = a and h(a)=4, which of the following must be true?
limx→a−h(x) ϵ R and limx→a+h(x) ϵ R
Given that
limx→a[3f(x)+g(x)]=6.........(i)
limx→a[2f(x)−g(x)]=4...........(ii)
(i)+(ii)⇒limx→a[5f(x)]=10⇒limx→af(x)=2⇒limx→ag(x)=0=g(a) (∵g(x) is continuous)
Since h(x).g(x) is continuous,
limx→ah(x).limx→ag(x)=h(a).g(a)⇒limx→ah(x)×0=4×0
This is true if limx→ah(x)≠∞.