1. **Problem statement:** Given the function $$f(x) = 2x - 5 - 4xe^{-0.5x}$$ defined on $$[0,+\infty[$$, determine $$\lim_{x \to +\infty} f(x)$$. 2. **Formula and rules:** To find the limit of $$f(x)$$ as $$x \to +\infty$$, analyze each term separately. Recall that $$\lim_{x \to +\infty} e^{-ax} = 0$$ for any positive constant $$a$$. 3. **Intermediate work:** $$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} \left(2x - 5 - 4xe^{-0.5x}\right) = \lim_{x \to +\infty} (2x - 5) - \lim_{x \to +\infty} 4xe^{-0.5x}$$ 4. The first limit is: $$\lim_{x \to +\infty} (2x - 5) = +\infty$$ 5. For the second limit, use the fact that exponential decay dominates polynomial growth: $$\lim_{x \to +\infty} 4xe^{-0.5x} = 0$$ 6. Therefore, $$\lim_{x \to +\infty} f(x) = +\infty$$ **Final answer:** $$\lim_{x \to +\infty} f(x) = +\infty$$