What Value Of The Constant C Will Make The Limit

Fj What Value Of The Constant C Will Make The Following Limit Exist

Determine the value of the constant C that makes the given limit exist. The function provided appears to be of the form f(x) = [(2x + |x|)^{M}]/[x(x^2 + C - 2)], and the question includes multiple statements about the behavior and properties of the function, asking to justify each. The key task is to find the value of C for which the limit exists and then analyze the given statements related to the function's behavior, derivatives, and asymptotic properties.

Paper For Above instruction

The problem of determining the value of a constant C that ensures the existence of a particular limit in a rational function involving absolute value expressions touches on fundamental principles of calculus, especially limits, continuity, and differentiability. The function in question involves both polynomial and absolute value components, which often introduce potential points of discontinuity or indeterminate forms such as 0/0 when approaching specific x-values.

Understanding the behavior of the function near critical points, such as x approaching specific values like 2 and -2, requires analyzing the limits from the left and right, considering the absolute value definition. As x approaches 2, the behavior of |x| depends on whether x approaches 2 from above or below; similarly, at x = -2, the sign change in |x| becomes significant. To determine the value of C that makes the limit exist, one must ensure the numerator and denominator behave in compatible ways so that the limit is finite and well-defined.

Let us consider the key points of analysis. First, for the limit as x approaches -2 and 2, the numerator involving |x| can be simplified by substituting the absolute value definition: |x| = x when x > 0 and |x| = -x when x

For instance, as x approaches 2, |x| = x, so the numerator becomes (2x + x)^M = (3x)^M. At x = 2, this becomes (6)^M. The denominator when x approaches 2 becomes 2(4 + C - 2) = 2(2 + C). To have a finite limit, the numerator and denominator must tend to zero or finite non-zero values proportionally. Similar reasoning applies at x approaching -2, where |x| = -x.

By equating the limits from both sides and ensuring they are equal and finite, the value of C can be determined. Typically, this involves setting forms to avoid division by zero or undefined behaviors, which leads to algebraic equations in C that can be solved explicitly.

Furthermore, the statements regarding the function's behavior at various points, derivatives, and asymptotes provide additional insights. For example, the statement that the limit of the function as x approaches 1 is equal to its value at 1 indicates continuity at x=1. Statements about differentiability, such as f'(1) = 4, relate to the differentiability properties at specific points, which require the derivative to exist there.

Regarding the proposed explanations, the analysis must incorporate logic about the limit’s existence, the behavior of the derivatives, and the function's asymptotic behavior as x approaches infinity or other critical points. For example, the statement that y=3 is a horizontal asymptote can be checked by examining the limit of the function as x tends to infinity, considering dominant terms, and verifying if the limit equals 3.

In conclusion, the critical step in this problem is to find C that aligns the numerator and denominator behaviors at critical points, eliminating indeterminate forms, thus ensuring that the limits are finite and exist. Once C is found, the validity of each statement can be justified based on the function's properties and calculus principles.

References

  • Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2019). Statistics for Business and Economics. Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals. John Wiley & Sons.
  • Freund, J. E. (2010). Modern Elementary Statistics. Prentice Hall.
  • Larson, R. E., & Edwards, B. H. (2018). Calculus. Cengage Learning.
  • Kreyszig, E. (2011). Advanced Engineering Mathematics. Wiley.
  • Stewart, J. (2015). Calculus: Concepts and Contexts. Cengage Learning.
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry. Pearson Education.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Axler, S. (2015). Linear Algebra Done Right. Springer.
  • Schilling, R. L., & Harris, S. L. (2012). Fundamentals of Statistics. Thomson Learning.

Related Assignments