Limit based on bounds

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Let $k\in(0,1)$ be fixed. If $\limsup_{x\to\infty}f(kx)\leq \epsilon$ and $\liminf_{x\to\infty}f(\frac{x}{k})\geq -\epsilon$ for all $\epsilon>0$ then is it possible to say $$\lim_{k\to\infty}f(x)=0.$$