Question

f(x) and g(x) are continuous functions such that ${lim}_{x\to a}\left[3f\right(x)+g(x\left)\right]=6and{lim}_{x\to a}\left[2f\right(x)-g(x\left)\right]=4$. Given that the function h(x).g(x) is continuous at x = a and h(a)=4, which of the following must be true?

• A. ${lim}_{x\to a^{-}}h\left(x\right)ϵR$ and ${lim}_{x\to a^{+}}h\left(x\right)ϵR$
• B. ${lim}_{x\to a}h\left(x\right)=0$
• C. ${lim}_{x\to a}h\left(x\right)=\infty$
• D. ${lim}_{x\to a}h\left(x\right)=4$

Answer 1

The correct option is A ${lim}_{x\to a^{-}}h\left(x\right)ϵR$ and ${lim}_{x\to a^{+}}h\left(x\right)ϵR$

Given that
${lim}_{x\to a}\left[3f\right(x)+g(x\left)\right]=6.........\left(i\right)$
${lim}_{x\to a}\left[2f\right(x)-g(x\left)\right]=\mathrm{4...........}\left(ii\right)$
$\left(i\right)+\left(ii\right)\Rightarrow {lim}_{x\to a}\left[5f\right(x\left)\right]=10\Rightarrow {lim}_{x\to a}f\left(x\right)=2\Rightarrow {lim}_{x\to a}g\left(x\right)=0=g\left(a\right)(\because g(x\left)iscontinuous\right)$
Since h(x).g(x) is continuous,
${lim}_{x\to a}h\left(x\right).{lim}_{x\to a}g\left(x\right)=h\left(a\right).g\left(a\right)\Rightarrow {lim}_{x\to a}h\left(x\right)\times 0=4\times 0$
This is true if ${lim}_{x\to a}h\left(x\right)\ne \infty$.