Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=1$ and $\int_0^{\frac{\pi}{3}} f(t) d t=0$.
Then which of the following statements is (are) TRUE?
Select ALL correct options:
A
The equation $f(x)-3 \cos 3 x=0$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
B
The equation $f(x)-3 \sin 3 x=-\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
C
$\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) d t}{1-e^{x^2}}=-1$
D
$\lim _{x \rightarrow 0} \frac{\sin x \int_0^x f(t) d t}{x^2}=-1$
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