Let $$ G(t)=\int_{0}^{t} g(x)\ln|x| \mathrm dx$$
Give conditions over $g(x)$ so that $G$
1.It is defined near $0$.
2.Be continuous in $t=0$.
3.Be derivable near $0$
4.Give the value of $$\lim_{t\to0} \frac{G(t)}{3t}$$
Questions about reasoning:
I have several questions about this. First of all, if a function (in this case G) is defined with an integral, can it not be be defined, in this case, in $0$. Is it an improper Integral
If i define g(x) so that $g(x)\ln|x|$ is continuous in t=0 that would mean(by theorem) that $g(x)\ln|x|$ can be integrated there and by, another theorem, that G is continuous ?
I don't know. Any help thinking this through ?
I would take $$g(x)=x$$.
Then $$G(t)=\ln|t|\frac{t^2}2-\frac{t^2}4$$
It can be easily seen that all requirements are met and the limit is $0$.