Convert an inequality into limiting equality.

Question

Given $f(x)/g(x) \lt 1.5/h(x)$ where all three functions are increasing and positive in nature. My question is, if I can deduce $\lim_{x \to \infty} [f(x)/g(x)]=1$ (then how if yes) or not .?

Answer 1

I'm afraid you can't deduce this. For example, suppose you have $f(x),g(x)$ for which your limit holds. For any $h(x)$ for which your inequality also holds, you have

$$\frac{f(x)}{g(x)}<\frac{1.5}{h(x)}.$$

But then consider $\bar f(x)=f(x)/2$. Since $f$ is positive, $\bar f<f$, so we must have

$$\frac{\bar f(x)}{g(x)}<\frac{1.5}{h(x)}$$

or, in other words, your same inequality holds. But now

$$\lim_{x\to\infty}\frac{\bar f}{g}=.5\lim_{x\to\infty}\frac{ f}{g}=.5.$$