suppose that the only thing we know about functions $f$ and $g$ is that $lim_{x\rightarrowinfty}f(x)=0$…

Explanation:

Step1: Analyze $\lim_{x\rightarrow\infty}(f(x)-g(x))$

We know $\lim_{x\rightarrow\infty}f(x) = 0$ and $\lim_{x\rightarrow\infty}g(x)=\infty$. Then $\lim_{x\rightarrow\infty}(f(x)-g(x))=0 - \infty=-\infty$. But this is an indeterminate - form in a sense of limit rules. It is not a well - defined real number limit.

Step2: Analyze $\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)}$

Since $\lim_{x\rightarrow\infty}f(x) = 0$ and $\lim_{x\rightarrow\infty}g(x)=\infty$, we have the form $\frac{\infty}{0}$, which is an indeterminate form and the limit does not exist as a real number.

Step3: Analyze $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}$

We use the quotient rule for limits. Given $\lim_{x\rightarrow\infty}f(x) = 0$ and $\lim_{x\rightarrow\infty}g(x)=\infty$. By the rule $\lim_{x\rightarrow a}\frac{u(x)}{v(x)}=\frac{\lim_{x\rightarrow a}u(x)}{\lim_{x\rightarrow a}v(x)}$ (when $\lim_{x\rightarrow a}v(x)\neq0$ in the non - extended real number sense, but in the context of limits to infinity), we have $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{0}{\infty}=0$.

Step4: Analyze $\lim_{x\rightarrow\infty}(f(x)\cdot g(x))$

This is in the $0\times\infty$ indeterminate form. We cannot determine its value without more information about the functions $f$ and $g$.

Step5: Analyze $\lim_{x\rightarrow\infty}(f(x)+g(x))$

Since $\lim_{x\rightarrow\infty}f(x) = 0$ and $\lim_{x\rightarrow\infty}g(x)=\infty$, we have $\lim_{x\rightarrow\infty}(f(x)+g(x))=0+\infty=\infty$. But this is not a real - valued limit.

Answer:

$\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}$