Răspuns:
Explicație pas cu pas:
[tex]\displaystyle\texxx{Consideram functia }f:\mathbb{R}\rightarrow \mathbb{R},f(x)=x^2\cdot e^{-x^2}\cdot \sin x\texttt{ si fie }G\texttt{ o }\\\texttt{primitiva a sa.}\\\lim_{x\to 0}\dfrac{1}{x^4}\int_0^xt^2\cdot e^{-t^2}\cdot \sin tdt=\lim_{x\to 0}\dfrac{\int_0^x t^2\cdot e^{-t^2}\cdot \sin tdt}{x^4}=\dfrac{G(x)-G(0)}{x^4}\stackrel{\frac{0}{0}}{=}\\\lim_{x\to 0}\dfrac{g(x)}{4x^3}=\lim_{x\to 0}\dfrac{x^2\cdot e^{-x^2}\cdot\sin x}{4x^3}=\lim_{x\to 0}\dfrac{e^{-x^2}\cdpt\sin x}{4x}=\boxed{\dfrac{1}{4}}[/tex]
