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Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$. Can we use upper limits or lower limits to replace the above limits? i.e. we only know $\limsup_{t\to 0}f(t)$ or $\liminf_{t\to0}f(t)$ exist. Can we get the following equality? $$\limsup_{t\to 0}f(t)=\limsup_{s\to\infty}sF(s), $$ $$\liminf_{t\to 0}f(t)=\liminf_{s\to\infty}sF(s)? $$ If it is correct, is there someone can give me some reference or if it is incorrect is there someone can give me a counterexample?

Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$. Can we use upper limits or lower limits to replace all the above limits? i.e. we only know $\limsup_{t\to 0}f(t)$ or $\liminf_{t\to0}f(t)$ exist. Can we get the following equality? $$\limsup_{t\to 0}f(t)=\limsup_{s\to\infty}sF(s), $$ $$\liminf_{t\to 0}f(t)=\liminf_{s\to\infty}sF(s)? $$ If it is correct, is there someone can give me some reference or if it is incorrect is there someone can give me a counterexample?

Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$. Can we use upper limits or lower limits to replace the above limits? i.e. Can we get the following equality? $$\limsup_{t\to 0}f(t)=\limsup_{s\to\infty}sF(s), $$ $$\liminf_{t\to 0}f(t)=\liminf_{s\to\infty}sF(s)? $$ If it is correct, is there someone can give me some reference or if it is incorrect is there someone can give me a counterexample?

Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$. Can we use upper limits or lower limits to replace the above limits? i.e. we only know $\limsup_{t\to 0}f(t)$ or $\liminf_{t\to0}f(t)$ exist. Can we get the following equality? $$\limsup_{t\to 0}f(t)=\limsup_{s\to\infty}sF(s), $$ $$\liminf_{t\to 0}f(t)=\liminf_{s\to\infty}sF(s)? $$ If it is correct, is there someone can give me some reference or if it is incorrect is there someone can give me a counterexample?