Let, $f:(0,1]\to (0,\infty)$ be an increasing function with $\lim_{x\to 0} f(x) = 0$ and $g:(0,1]\to (0,\infty)$ be a decreasing function with $\lim_{x\to 0} g(x) = +\infty$. Also, if $\lim_{x\to 0} f(x)g(x) $ exists. Then, can it be shown that the minimum value of function $fg$ will exist at $x=0$, if we redefine $fg$ at $x=0$.
If it is not true, then is there any suitable example for it?
Required counter example, $f(x)=x,g(x)=\frac{1}{x}-x+\frac{1}{2}$, thanks to @geetha290krm