简单的导数构造
2018.05.20
\(f\left( x \right) \text{在}\left[ a,+\infty \right] \text{可导，}g\left( x \right) \text{在}\left[ a,+\infty \right] \text{连续且恒大于0，}\int_a^{+\infty}{g\left( x \right) dx}\text{发散。已知}\lim_{x\rightarrow \infty}\left[ f\left( x \right) +\frac{f'\left( x \right)}{g\left( x \right)} \right] =0,\text{证：}f\left( x \right) \rightarrow 0\left( x\rightarrow +\infty \right) .\)
\(solution:\)
\(G\left( x \right) =\int_a^x{g\left( t \right) dt},\lim_{x\rightarrow \infty}\left[ \frac{f\left( x \right) G'\left( x \right) +f'\left( x \right)}{G'\left( x \right)} \right] =\lim_{x\rightarrow \infty}\left[ \frac{\left( f\left( x \right) G'\left( x \right) +f'\left( x \right) \right) e^{G\left( x \right)}}{G'\left( x \right) e^{G\left( x \right)}} \right] =\lim_{x\rightarrow \infty}\left[ \frac{\left( f\left( x \right) e^{G\left( x \right)} \right) '}{\left( e^{G\left( x \right)} \right) '} \right] =\lim_{x\rightarrow \infty}f\left( x \right) =0.\)
