1. **Problem statement:** Calculate the limits of the function $$f(x) = (-x^3 + 2x^2) e^{-x+1}$$ as $x \to +\infty$ and $x \to -\infty$. 2. **Formula and rules:** To find limits involving polynomials multiplied by exponentials, analyze the dominant terms: - As $x \to +\infty$, exponential decay $e^{-x}$ tends to 0 faster than any polynomial grows. - As $x \to -\infty$, exponential $e^{-x}$ grows exponentially since $-x \to +\infty$. 3. **Calculate limit as $x \to +\infty$:** $$\lim_{x \to +\infty} (-x^3 + 2x^2) e^{-x+1} = e^{1} \lim_{x \to +\infty} (-x^3 + 2x^2) e^{-x}$$ Since $e^{-x}$ decays faster than any polynomial grows, $$\lim_{x \to +\infty} (-x^3 + 2x^2) e^{-x} = 0$$ Therefore, $$\lim_{x \to +\infty} f(x) = 0$$ 4. **Calculate limit as $x \to -\infty$:** Rewrite exponential: $$e^{-x+1} = e^{1} e^{-x} = e^{1} e^{|x|}$$ As $x \to -\infty$, $|x| \to +\infty$, so $e^{|x|} \to +\infty$. The polynomial term: $$-x^3 + 2x^2 = -(-|x|)^3 + 2(-|x|)^2 = |x|^3 + 2|x|^2$$ which grows positively large. Thus, $$f(x) \approx (|x|^3 + 2|x|^2) e^{1} e^{|x|} \to +\infty$$ 5. **Final answers:** - $$\lim_{x \to +\infty} f(x) = 0$$ - $$\lim_{x \to -\infty} f(x) = +\infty$$ This shows the function tends to zero at positive infinity and grows without bound at negative infinity.