Sequences: Group 4 - Basic Limits and Continuity | Problems #6 and #7.

In this lesson, we explore Group 4 — Basic Limits and Continuity, focusing on Problems #6 and #7 from Calculus II, Section 8.1: Sequences. We solve each problem step by step, explaining the mathematical rules, formulas, and reasoning necessary to determine whether a sequence converges or diverges. Problem #6 — Trigonometric Sequences and Continuity aₙ = sin(1/(n + 1)) We examine the limit of the inner expression before applying the continuity of the sine function. lim(n approaching ∞) 1/(n + 1) = 0 Since sine is continuous at zero: lim(n approaching ∞) sin(1/(n + 1)) = sin(0) = 0 Conclusion: The sequence converges to 0. Problem #7 — Exponential Sequences and Negative Exponents aₙ = e⁻ⁿ We apply the negative exponent rule: e⁻ⁿ = 1/eⁿ = (1/e)ⁿ Because e greater then 1, the denominator eⁿ increases without bound as n approaches infinity. Therefore: lim(n approching ∞) e⁻ⁿ = 0 Conclusion: The sequence converges to 0. Mathematical Concepts Covered: Understanding sequences and their limits Convergence versus divergence Evaluating limits as n approaches infinity Continuity of trigonometric functions Evaluating sine limits Applying negative exponent rules Exponential growth and decay Recognizing geometric sequences Interpreting mathematical graphs and sequence behavior Identifying the most efficient solution strategy The lesson includes mathematical explanations, formulas, numerical examples, graphical illustrations, and a final recap to reinforce understanding. Designed for students learning these concepts for the first time and those preparing for calculus quizzes and examinations.