Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.
$x^3[f(x+1)-f(x-1)]=1$
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functional-equations
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3If $g(x)=f(x)-f(x-1)$, then $g(x+1)+g(x)=1/x^3$. It follows that *if your limit exists*, then its value is zero. But there is no reason for it to exists: you can construct pretty wild solutions by starting from an arbitrary function on an interval of length $2$ and using the equation to continue it. – 2011-01-19
1 Answers
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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 $l=0$.