1. **Problem Statement:** We need to evaluate the latency for four Azure Availability Zones using given limit formulas as $x \to 6$ and rank them by latency. Then, evaluate two additional limits as $x \to \infty$ to assess performance improvements. 2. **Recall Limit Rules:** - If direct substitution in a limit yields a finite number, that is the limit. - If substitution yields an indeterminate form like $\frac{0}{0}$, use algebraic simplification or L'Hôpital's Rule. - For limits at infinity, compare degrees of polynomials in numerator and denominator. --- ### 5.1 Latency for Zones 1 to 4 (as $x \to 6$): **Zone 1:** $$\lim_{x \to 6} (3x^3 - 6x^2 - 400)$$ Substitute $x=6$: $$3(6)^3 - 6(6)^2 - 400 = 3(216) - 6(36) - 400 = 648 - 216 - 400 = 32$$ **Zone 2:** $$\lim_{x \to 6} \frac{x^3 - 36}{x^2 - 6}$$ Substitute $x=6$: Numerator: $6^3 - 36 = 216 - 36 = 180$ Denominator: $6^2 - 6 = 36 - 6 = 30$ Limit = $\frac{180}{30} = 6$ **Zone 3:** $$\lim_{x \to 6} \frac{x + 6}{x^3 - 36}$$ Substitute $x=6$: Numerator: $6 + 6 = 12$ Denominator: $6^3 - 36 = 216 - 36 = 180$ Limit = $\frac{12}{180} = \frac{1}{15} \approx 0.0667$ **Zone 4:** $$\lim_{x \to 6} \frac{x^2 - x^2}{2^2 - 6} = \lim_{x \to 6} \frac{0}{4 - 6} = \frac{0}{-2} = 0$$ **Ranking by latency (lowest is best):** Zone 4 (0), Zone 3 (~0.0667), Zone 2 (6), Zone 1 (32) **Best zone to store data:** Zone 4 due to lowest latency. --- ### 5.2 Evaluate limit for performance improvement: $$\lim_{x \to \infty} \frac{2x^2 - 3x + 6}{8x^2 + x + 6}$$ Divide numerator and denominator by $x^2$ (highest power): $$\lim_{x \to \infty} \frac{2 - \frac{3}{x} + \frac{6}{x^2}}{8 + \frac{1}{x} + \frac{6}{x^2}} = \frac{2}{8} = \frac{1}{4} = 0.25$$ Since this limit is finite and relatively low, it suggests improved performance compared to higher latencies. --- ### 5.3 Evaluate new availability zone latency: $$\lim_{x \to \infty} \frac{x + x^2}{2^2 + x^2} = \lim_{x \to \infty} \frac{x^2 + x}{4 + x^2}$$ Divide numerator and denominator by $x^2$: $$\lim_{x \to \infty} \frac{1 + \frac{1}{x}}{\frac{4}{x^2} + 1} = \frac{1 + 0}{0 + 1} = 1$$ This limit is 1, which is higher than 0.25 from 5.2, so it does not improve latency as much. --- **Final conclusions:** - Best zone at $x=6$ is Zone 4. - The limit in 5.2 (0.25) indicates better performance than 5.3 (1). - Use the availability zone from 5.2 for improved cloud storage latency.