Continuous functions such that $f/g \to 1$ but $f-g \to +\infty$

Question

Do you have an example of continuous functions $f,g : \Bbb R_{>0} \to \Bbb R$ such that $$f(x)/g(x) \to 1 \text{ as } x \to +\infty \qquad\qquad f(x)-g(x) \to +\infty \text{ as } x \to +\infty$$

I had in mind functions like $f(x)=x$ and $g(x)=x+2$ or $g(x)=x+\sin(x)$. The first condition is satisfied but the second is only having a finite limit, or having no limit at all.

Answer 1

foe example : $$f(x)=(x+1)^2\\g(x)=x^2\\ \lim_{x\to \infty }\dfrac{f(x)}{g(x)}=1\\ \lim_{x\to \infty }f(x)-g(x)=\lim_{x\to \infty }(x^2+2x+1-x^2)\to \infty$$

Answer 2

What about $f(x)=x^2+x$, $g(x)=x^2$?