11. ¿cual de las siguientes expresiones define correctamente la transformada de Laplace de una \( f(t) \) ? (A) \( \mathcal{L}\{f(t)\}=\int_{0}^{\infty} f(t) e^{s t} d t \) (B) \( \mathcal{L}\{f(t)\}=\int_{-\infty}^{\infty} f(t) e^{-s t} d t \) (C) \( \mathcal{L}\{f(t)\}=\int_{0}^{\infty} f(t) e^{-s t} d t \) (D) \( \mathcal{L}\{f(t)\}=\int_{-\infty}^{\infty} f(t) e^{x t} d t \) (E) \( \mathcal{L}\{f(t)\}=\int_{-\infty}^{\infty} f(t) t^{s} d t \)

Calcula \( \lim _{x \rightarrow-\infty}\left(\frac{3 x^{2}-5 x+8}{x^{3}+7 x^{2}-5 x}\right) \)

Calcula \[ \operatorname{Lim}_{x \rightarrow \infty}\left(\frac{8 x^{2}-3 x+9}{2 x^{2}+7 x-10}\right) \]

2. Use the reduction formula for \( \mathrm{I}_{\mathrm{n}}=\int_{0}^{\pi / 2} \cos ^{n} x d x \), to show that \[ I_{(n-4)}=\left(\frac{n-5}{n-4}\right) I_{(n-6)} \quad(n \geq 2) \]

46. \( \int_{0}^{2} x e^{x} d x \) ni hisoblang. \( \begin{array}{llll}\text { A) } e & \text { B) } e^{2}-1 & \text { C) } e^{2}+1\end{array} \)

Calcula \[ \lim _{x \rightarrow-\infty}\left(\frac{x^{3}-3 x^{2}+8}{x^{2}+4 x-10}\right) \]