CSU Webwork MATH 160 Assignment M-FA-1.2 Due 09/02/22

CSU Webwork MATH 160 WeBWorK assignment M-FA-1.2 due 09/02/2016 at 11:59pm MDT

Evaluate the following limits, determine if they exist, and provide the appropriate value or expression. For limits approaching infinity or negative infinity, use I for positive infinity and -I for negative infinity. If a limit does not exist, write DNE. When simplifying complex expressions involving absolute values, rewrite them without absolute value bars to facilitate direct evaluation. For limits involving functions with piecewise definitions, analyze the left-hand and right-hand limits separately and compare to determine the overall limit.

Sketching functions and finding their limits at specific points are also required, including identifying whether these limits exist. Additionally, find the value of constants that make functions continuous across specified points by equating left-hand and right-hand limits at those points.

Paper For Above instruction

The evaluation of limits is fundamental in calculus, serving as the cornerstone for understanding continuity, derivatives, and integrals. Limits describe the behavior of functions as their inputs approach specific values, and correctly evaluating or determining their existence is crucial for analyzing and interpreting these functions in various mathematical contexts.

Analysis of Various Limits and Continuity

Several of the given problems involve evaluating limits at specific points, some of which involve indeterminate forms, infinite limits, or piecewise functions. For instance, the limit as y approaches 1 of (y³ - y)/(y² - 1) can be simplified using algebraic techniques, such as factoring numerator and denominator or applying L'Hôpital's Rule when appropriate. In this case, factorization reveals the indeterminate form 0/0, allowing simplification to (y(y²-1))/(y²-1), leading to the limit as y approaches 1 of y, which is 1.

Similarly, limits involving quadratic expressions, such as lim t→ -7 (t² - 49) / (-2t² - 12t +14), require factoring numerator and denominator to identify cancellations and evaluate the limit or infer that it DNE if the limits from the left and right do not agree.

Limits involving radicals, such as lim x→1 (√x - x²) / (1 - √x), often involve rationalizing the numerator or denominator to simplify the expression to a form where direct substitution yields the limit. For instance, rationalizing the numerator by multiplying numerator and denominator by (√x + x²) may clarify whether the limit exists and evaluate its value.

Piecewise functions require examination of left and right limits separately. For example, analyzing the function f(x), defined differently on (−∞, -5], (-5,9), and beyond, involves approaching the point of interest from both sides and determining whether the function values approach a common limit, thereby establishing the existence of the overall limit at that point.

In the context of these problems, understanding how to manipulate algebraic expressions, apply special limits, and understand the behavior of radical functions is necessary for accurate evaluation. For limits involving absolute values, rewriting expressions without absolute value bars—by analyzing the sign of the expression inside—is critical for simplifying and evaluating the limits properly.

Furthermore, problems involving the determination of constant values for continuity, such as ensuring the function is continuous at a specific point, revolve around setting the left and right limits equal and solving for the constant. This process guarantees that the function does not have a break or jump at the specified point, which is essential for many applications in calculus and analysis.

Overall, mastering limit evaluation, especially involving indeterminate forms, radicals, piecewise functions, and parameters, is vital in the broader context of calculus, providing the foundation for the subsequent study of derivatives, integrals, and the behavior of functions near points of interest.

References

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• Larson, R., & Edwards, B. (2019). Calculus (11th ed.). Cengage Learning.
• Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
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