determine whether the following integral converges or diverges. \n∫0,∞ dx / (e^x + 3x + 8)\nas x approaches…

determine whether the following integral converges or diverges. \n∫0,∞ dx / (e^x + 3x + 8)\nas x approaches ∞, which of the following is true about the behavior of the integrand 1 / (e^x + 3x + 8)?\na. 1 / (e^x + 3x + 8) < e^x / (e^2x + 3x + 8)\nb. 1 / (e^x + 3x + 8) < 1 / e^x\nc. 1 / (e^x + 3x + 8) > 1 / e^2x\nd. 1 / (e^x + 3x + 8) > 1 / e^x

determine whether the following integral converges or diverges. \n∫0,∞ dx / (e^x + 3x + 8)\nas x approaches ∞, which of the following is true about the behavior of the integrand 1 / (e^x + 3x + 8)?\na. 1 / (e^x + 3x + 8) < e^x / (e^2x + 3x + 8)\nb. 1 / (e^x + 3x + 8) < 1 / e^x\nc. 1 / (e^x + 3x + 8) > 1 / e^2x\nd. 1 / (e^x + 3x + 8) > 1 / e^x

Answer

Explanation:

Step1: Analyze the growth - rate of functions

As (x\to\infty), the exponential function (e^{x}) dominates the linear function (3x) and the constant function (8). So, (e^{x}+3x + 8\approx e^{x}) as (x\to\infty). Since (e^{x}+3x + 8>e^{x}) for (x>0), then (\frac{1}{e^{x}+3x + 8}<\frac{1}{e^{x}}) for (x > 0).

Answer:

B. (\frac{1}{e^{x}+3x + 8}<\frac{1}{e^{x}})