Definition of $\limsup_{x\to 0^+}f(x)$

Question

By definition of limit superior, we have $\limsup_{n\to \infty}x_n=\lim_{n\to \infty}(\sup_{m\geq n}x_m)$. Whats is the definition of $\limsup_{x\to 0^+}f(x)$?

Answer 1

The definition of $\limsup_{x\to0^+}f(x)$ is $$ \lim_{x\to 0^+}\left(\sup_{0<y<x}f(y)\right) $$ Can you see how similar this is to your $\limsup_{n\to\infty}x_n$ definition?