Question

If \( \lim_{t \to \infty} \left( \int_0^{1} \left( 3x + 5 \right)^t dx \right) = \frac{\alpha}{5e} \left( \frac{8}{5} \right)^{\frac{3}{2}}, \) then \( \alpha \) is equal to ____ :

Hint:

For such problems involving limits, observe the behavior of exponential terms and simplify using the highest powers.

Answer 1

Correct Answer: 64

Step 1: Using the formula for the limit of the given integral. We have the integral: \[ L = \int_0^1 \frac{(3x + 5)^t}{t (3(t + 1))} dx \] As \( t \to \infty \), the exponential terms dominate. Therefore, we calculate: \[ L = \lim_{t \to \infty} e^{8t} (3t + 5t - 3t) \] \[ = e^{8t} n8 - 5n5 - 3 \] Finally comparing values, we find: \[ \alpha = 64 \]