Limsup and liminf of a function

Question

Let $k\in(0,1)$ be fixed and $L\in \mathbb{R}$ is finite.

If $\limsup_{x\to\infty}f(kx)=L$ and $\liminf_{x\to\infty}f(\frac{x}{k})=L$ then is it possible to say $$\lim_{x\to\infty}f(x)=L.$$

Answer 1

Is it true that

$\limsup_{x\to\infty}f(kx)=L \ \Rightarrow \limsup_{x\to\infty}f(x)=L$

and

$\liminf_{x\to\infty}f(\frac{x}{k})=L \Rightarrow \liminf_{x\to\infty}f(x)=L \ \ \ $?

And if so, does that give you what you want?

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Added

Write out the definitions and follow your nose. To get started, $\limsup_{x\to\infty}f(kx)=L$ means that $\lim_{x\to\infty} \sup\{ f(kx) \} = L$ i.e., for all $\epsilon > 0$, there exists an $M > 0$ such that

$$ | \sup\{ f(kx) : x > M \} - L \ | < \epsilon$$

That is,

$$ | \sup\{ f(x) : x > kM \} - L \ | < \epsilon$$

In other words, for arbitrary $\epsilon > 0$, there exists a constant $M'$, namely $M' = kM$, such that $$ | \sup\{ f(x) : x > M' \} - L \ | < \epsilon$$

That is, $\limsup_{x\to\infty}f(x)=L$