joybangla wrote: Does there exist points $A,B$ on the curve $y=x^3$ and on $y=x^3+|x|+1$ respectively such that distance between $A,B$ is less than $\frac{1}{100}$ ? $\lim_{x\to+\infty}\left(\sqrt[3]{x^3+x+1}-x\right)=0$. So yes : choose $t>0$ such that $\sqrt[3]{t^3+t+1}-t\in(0,\frac 1{100})$ and $A(\sqrt[3]{t^3+t+1},t^3+t+1)$ and $B(t,t^3+t+1)$