$x^3[f(x+1)-f(x-1)]=1$

Question

Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.

Answer 1

if $\lim_{x\to\infty}(f(x)-f(x-1))$ exists and it's equal to $l$, then also $\lim_{x\to\infty}(f(x+1)-f(x))=l$. So $$ 2l=\lim_{x\to\infty}( f(x+1)-f(x-1))=\lim_{x\to\infty}\frac{1}{x^3}=0, $$ hence $l=0$.