this question is designed to be answered without a calculator.\n$lim_{x\rightarrowinfty}left(\frac{20x^{4}-1}…

Explanation:

Step1: Analyze growth - rate of functions

As (x\to\infty), the exponential function (y = e^{x}) grows much faster than any polynomial function. The polynomial function is (20x^{4}-1) and the exponential - related function is (4e^{x}-1).

Step2: Use the limit property

We know that (\lim_{x\to\infty}\frac{20x^{4}-1}{4e^{x}-1}). Divide both the numerator and denominator by (e^{x}): (\lim_{x\to\infty}\frac{\frac{20x^{4}}{e^{x}}-\frac{1}{e^{x}}}{4 - \frac{1}{e^{x}}}). We know that (\lim_{x\to\infty}\frac{x^{n}}{e^{x}} = 0) for any positive integer (n) and (\lim_{x\to\infty}\frac{1}{e^{x}}=0). So (\lim_{x\to\infty}\frac{20x^{4}}{e^{x}} = 0) and (\lim_{x\to\infty}\frac{1}{e^{x}} = 0).

Step3: Calculate the limit

(\lim_{x\to\infty}\frac{\frac{20x^{4}}{e^{x}}-\frac{1}{e^{x}}}{4 - \frac{1}{e^{x}}}=\frac{0 - 0}{4-0}=0).

Answer:

A. 0