5.2 解答

5.2

1.

(a)

$$\lim_{t \rightarrow \infty}\frac{\sin{3t}}{e^{at}} = 0 \ (a > 0) \ \ \framebox{終}$$

(b)

$$\lim_{t \rightarrow \infty}\frac{e^{100t}}{e^{at}} = 0 \ (a > 100) \ \ \ \framebox{終}$$

(c)

$$\lim_{t \rightarrow \infty}\frac{\log{(1+t)}}{e^{at}} = \lim_{t \rightarrow \infty}\frac{\frac{1}{1+t}}{ae^{at}} = 0 \ (a > 0) \ \ \ \framebox{終}$$

(d)

$$\lim_{t \rightarrow \infty}\frac{t^{3}e^{7t}}{e^{at}} = \lim_{t \rightarrow \infty}\frac{t^{3}}{e^{(a-7)t}} = \lim_{t \rightarrow \infty}\frac{3t^{2}}{(a-7)e^{(a-7)t}}$$

$$= \cdots = \lim_{t \rightarrow \infty}\frac{6}{(a-7)^{3}e^{(a-7)t}} = 0 \ (a > 7) \ \ \framebox{終}$$