Definition of $\limsup$

Question

please tel me what is the definition of $$\limsup_{|u|\rightarrow\infty}\frac{2F(t,u)}{|u|^2}<\lambda$$ using $\varepsilon$

Please Thank you

Answer 1

By definition $\large\lim\sup_{|u|\to\infty}\frac{2F(t,u)}{|u|^2}=\lim_{|u|\to\infty}\sup_{|v|\ge|u|}\frac{2F(t,v)}{|v|^2}$

With $\varepsilon$:

$\forall \varepsilon\gt 0\,,\exists r\in \Bbb R^+_0:\frac{2F(t,u)}{|u|^2}\lt \lambda+\varepsilon,\forall u:|u|\gt r$.