\lim_{t o -1} E(t) = \lim_{t o -1} (t + 3) = -1 + 3 = 2

["Understanding Limits in Calculus: Evaluating limₜₒ₋₁ E(t) = limₜₒ₋₁ (t + 3) = 2", "In the study of calculus, limits are fundamental tools that help us analyze how functions behave as inputs approach specific values. Today, we explore a clear and instructive example:\n[\n\lim_{t \ o -1} E(t) = \lim_{t \ o -1} (t + 3) = 2\n]\nThis expression serves as a practical introduction to how limits work when a function approaches ( t = -1 ), and why careful substitution often simplifies evaluation.", "### What Does the Limit Mean?", "The notation\n[\n\lim_{t \ o -1} (t + 3)\n]\nmeans we are examining the behavior of the function ( E(t) = t + 3 ) as ( t ) gets arbitrarily close to ( -1 ), from both the left and the right.", "### Evaluating the Limit Directly", "For continuous functions—such as linear expressions—evaluating a limit as ( t ) approaches a value simply involves plugging in that value directly:\n[\n\lim_{t \ o -1} (t + 3) = (-1) + 3 = 2\n]\nThis direct substitution is valid because ( f(t) = t + 3 ) is continuous at ( t = -1 ), meaning there are no jumps, breaks, or discontinuities at this point.", "### Breaking the Limit: Why ( t \ o -1 )?", "The expression ( \lim_{t \ o -1} ) emphasizes the approach to ( t = -1 ), not the value at ( t = -1 ). Even though ( E(-1) = 2 ), the limit describes the value the function nears as ( t ) gets closer and closer to ( -1 ). Because ( E(t) ) is smooth and defined everywhere, the limit equals the actual function value.", "### Breaking It Down with Limits: Step-by-Step", "To reinforce the concept:\n1. Let ( t \ o -1 ), so ( t ) approaches ( -1 ) but never actually equals ( -1 ).\n2. Consider a sequence ( t_n = -1 + \frac{1}{n} ) as ( n \ o \infty ): each term gets closer to ( -1 ) from the right.\n Then:\n [\n \lim_{n \ o \infty} E(t_n) = \lim_{n \ o \infty} \left( (-1 + \frac{1}{n}) + 3 \right) = \lim_{n \ o \infty} \left( 2 + \frac{1}{n} \right) = 2\n ]\n3. Similarly, using a left-moving sequence ( t_n = -1 - \frac{1}{n} ):\n [\n \lim_{n \ o \infty} E(t_n) = \lim_{n \ o \infty} \left( (-1 - \frac{1}{n}) + 3 \right) = \lim_{n \ o \infty} \left( 2 - \frac{1}{n} \right) = 2\n ]\nBoth one-sided limits agree, confirming the overall limit.", "### Real-Wandomucker: Why This Matters", "Understanding limits like ( \lim_{t \ o -1} (t + 3) = 2 ) forms the foundation for more complex calculus topics: derivatives, integrals, and function continuity. Recognizing when direct substitution works avoids unnecessary computation and clarifies function behavior near critical points.", "### Final Result", "[\n\lim_{t \ o -1} E(t) = \lim_{t \ o -1} (t + 3) = 2\n]\nThis simple limit illustrates the core principle that for continuous functions, the limit approaching a point equals the function’s value at that point—here, ( E(-1) = 2 ).", "---", "Keywords: limit calculus, calculating limits, limit as t approaches -1, evaluate lim (t + 3), continuous functions, limit by substitution, one-sided limits, calculus fundamentals.\nMeta Description: Learn how to evaluate the limit limₜₒ₋₁ (t + 3) as t approaches -1 using direct substitution. A clear example showing why continuity allows easy limit calculation and how approaching values defines limit behavior."]