Page 1 Of 3 Math 233 Unit 1 Limits Individual Project Assign

Page 1 Of 3math233 Unit 1 Limitsindividual Project Assignment Versio

Using a graphing utility from the Internet or Excel, graph the following functions. Based on the graphs, estimate the given limit. Make sure to include the graphs in your answer form, and explain how you found your limit estimates. a. limₓ→∞ (x + 1) b. limₓ→∞ (x² + 1)/x. Find the limit (if it exists) of the following functions by completing the given tables. Round your answers to the nearest ten-thousandths. a. Let F(x) = x + 1. Find limₓ→1 F(x). x: 0.9, 0.99, 0.001, 1.01, 1.1 F(x): [values needed] b. Let G(x) = 5(x − 2)². Find limₓ→2 G(x). x: 1.9, 1.99, 1.001, 2.01, 2.1 G(x): [values needed] 3. Answer the following questions thoroughly based on the given graph of f(x). a. Is f(x) continuous at x = −1? b. Is f(x) continuous at x = 2? c. Is f(x) continuous at x = 4? 4. Let ðan(n) = (1 + n)^n. The limit of this function as n approaches 0 is a value that is very useful in some business applications. a. Complete the table below by calculating A(n), using the given values of n. Round your answer to the nearest ten-thousandths. n: −0.1, −0.01, −0.001, −0.0001, 0.0001, 0.001, 0.01, 0.1. A(n): [values needed] b. Based on the table, estimate the following values: i. limₙ→0− ðan(n) ii. limₙ→0+ ðan(n) iii. limₙ→0 ðan(n) 5. The cost, C (in millions of dollars) for a software company to seize x% of an illegal version of a gaming software that they developed is modeled by the following function: C(x) = x * 50 − 0.5x, 0 ≤ x

Paper For Above instruction

The comprehensive analysis of limits, functions, and their applications within real-world scenarios constitutes an essential component of calculus studies. This paper addresses multiple facets, including graphing for limit estimation, table analysis for function limits, continuity questions, exponential limits relevant to business, cost functions, and average cost analysis. Each is explored thoroughly, integrating mathematical reasoning with contextual interpretation to demonstrate understanding and practical application.

Graphing Functions and Estimating Limits

Graphing plays a crucial role in understanding the behavior of functions as they approach specific points or tend towards infinity. For the functions in question, graphing utilities such as Desmos or Excel provide visual representations that assist in estimating limits. For example, considering the limits as x approaches infinity of the functions x + 1 and (x² + 1)/x reveals important asymptotic behavior.

Graphing x + 1 shows a straight line with no bounds as x increases; therefore, limₓ→∞ (x + 1) is infinite. Similarly, graphing (x² + 1)/x indicates a quadratic over linear behavior, which simplifies to approximately x as x becomes large, implying the limit tends toward infinity. These visual aids reinforce the algebraic reasoning and enhance comprehension of asymptotic trends.

Tabular Analysis and Limit Calculation

Constructing value tables for functions like F(x) = x + 1 and G(x) = 5(x − 2)² enables precise numerical estimation of limits at specific points. By calculating function values at points approaching the target from both sides, one observes that as x approaches 1, F(x) approaches 2, confirming limₓ→1 F(x) = 2.

Similarly, for G(x), as x approaches 2, function values such as 1.9 and 2.01 approximate the limit, which is 0 at x=2, affirming the function's continuity at that point given the limit equals the function value there.

Continuity at Specific Points

Evaluating the continuity of a function at points like x = -1, 2, and 4 requires analyzing the limits from the left and right as well as the function's value at these points. If the limits from both sides equal the function value, the function is continuous there.

For instance, if f(x) at x = -1, 2, and 4 matches the limits approaching those points, the sense of continuity is confirmed. Discontinuities occur if any of these conditions are not met, such as a jump or removable discontinuity.

Limit of Exponential-Like Function and Business Applications

The sequence ðan(n) = (1 + n)^n is significant in modeling exponential growth or decay. Calculating A(n) for various n values reveals convergence behavior, with the classic limit limₙ→0+ (1 + n)^n being e, the base of natural logarithms.

Estimations for limits approaching zero from both sides show that limₙ→0− and limₙ→0+ are both close to e, confirming the known mathematical result that limₙ→0+ (1 + n)^n = e.

Cost Function and Business Scenario Interpretation

The cost function C(x) = x * 50 − 0.5x models expenses in seizing illegal software versions. By choosing a value of M between 20 and 120, and calculating costs at specific seizure rates, business decisions can be informed. The limit limₓ→100− C(x) indicates the maximum cost as seizure approaches complete (100%) control. If the function grows unbounded, it warns of escalating expenses—critical for resource allocation.

Cost Analysis and Average Costs

The total cost function C(x) = 80x + 30,000 represents fixed and variable expenses tied to production volume, which allows calculating the average cost per unit as A(x) = C(x) / x = 80 + 30000 / x. As production scales up, the average cost approaches 80, illustrating economies of scale.

Conclusion

The exploration of limits, continuity, functions, and their real-world applications underpin advanced calculus and operational decision-making. Accurate graphing, thorough table analysis, and contextual interpretation afford a comprehensive understanding. These mathematical tools are integral in fields such as economics, engineering, and business strategy, illustrating their practical relevance beyond abstract theory.

References

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• Mathematics Department, Khan Academy. Limits and Continuity. https://www.khanacademy.org/math/ap-calculus-ab