Given $f, g :\Bbb R\rightarrow\Bbb R$ be continuous functions whose graphs do not intersect. Then for which function below the graph lies entirely on one side of the $x$-axis
(1). $f$
(2). $g+f$
(3).$ g-f$
(4). $g.f$
I take an example $f(x)= logx$ and $g(x)=e^x$ then the graph of both functions do not Intersect.Let's discuss the options:(1) $f(x)= logx$ whose graph intersect $X$- axis at $x=1$.
(2) $(g+f)(x) = log( x) + e^x$$\implies lim_{x\to\infty}(logx+e^x)=\infty$
So $(f+g)(x)$ do not intersect positive $x$-axis,hence this option is false.
(3) $(g-f)(x )= (e^x - logx)$now
$\implies lim_{x\to\infty}(logx-e^x)=?$Now I am unable to evaluate this limit and solve this question completely and also please verify if the other solutions are correct.