Homework Help Overview
The discussion revolves around the limits of a piecewise function as \( x \) approaches 2 from the left and right. The original poster is attempting to prove that \( \lim_{x \to 2^{-}} f(x) = 10 \) and \( \lim_{x \to 2^{+}} f(x) = 10 \) using epsilon-delta definitions, specifically focusing on the functions \( f(x) = x^3 + 2 \) for \( x \to 2^{-} \) and \( f(x) = x^2 + 6 \) for \( x \to 2^{+} \).
Discussion Character
- Exploratory, Assumption checking, Mathematical reasoning
Approaches and Questions Raised
- The original poster outlines their approach to proving the limits using epsilon-delta definitions but questions how to determine the minimum delta values without a calculator. Participants discuss the necessity of defining the function and the implications of having separate deltas for one-sided limits.
Discussion Status
Participants are actively engaging with the original poster's attempts, providing guidance on the necessity of establishing a positive delta and discussing the implications of using minimum values for delta. There is a recognition of the need to show that such a delta exists without needing to calculate specific values.
Contextual Notes
There is a focus on ensuring that the definitions of the functions are clear, and participants emphasize the importance of maintaining the absolute values in the delta expressions. The discussion also highlights the constraints of the epsilon values being less than specific thresholds.