Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation $\frac{dy}{dx}=(2+5y)(5y-2)$, then the value of $\lim_{x\to-\infty} f(x)$ is ______.

$\dfrac{dy}{dx} = (2+5y)(5y-2) = 25y^{2} - 4$

$f(0) = 0$

$\lim_{x\to-\infty} f(x) = ?$

$1.\ \text{separate variables and integrate}$

$2.\ \text{use } f(0)=0 \text{ to fix } C$

$3.\ \text{take the limit } x\to-\infty$

$\dfrac{dy}{25y^2 - 4} = dx$

$\dfrac{1}{(5y-2)(5y+2)} = \dfrac{1}{4}\left(\dfrac{1}{5y-2} - \dfrac{1}{5y+2}\right)$

$\dfrac{1}{20}\ln\left|\dfrac{5y-2}{5y+2}\right| = x + C$

$x=0,\ y=0: \ \dfrac{1}{20}\ln\left|\dfrac{-2}{2}\right| = 0 = C \Rightarrow C = 0$

$\Rightarrow \left|\dfrac{5y-2}{5y+2}\right| = e^{20x}$

$x\to-\infty: \ e^{20x} \to 0 \Rightarrow \dfrac{5y-2}{5y+2} \to 0$

$\Rightarrow 5y - 2 \to 0 \Rightarrow y \to \dfrac{2}{5}$

$\lim_{x\to-\infty} f(x) = \dfrac{2}{5}$