How Can Redefining a Function’s Value Solve Your Limit Problems

Redefining a function’s value is a useful technique in mathematics for addressing specific issues with a function’s behavior at certain points. It’s a tool often utilized to enhance the function’s properties, making it more suitable for analysis and application in various mathematical contexts, particularly in calculus.

Understanding Redefining Function’s Value

1. Purpose: The primary goal of redefining a function’s value is to modify its behavior at certain points. This is usually done to:

• Remove discontinuities.
• Extend the domain of the function.
• Ensure certain properties like continuity or differentiability.

1. Removable Discontinuities: A common scenario for redefining a function involves removable discontinuities, where the function is not defined or behaves differently at a point compared to its immediate surroundings.
2. Continuous Extension: By redefining the value of a function at specific points, a continuous extension of the function can be created. This is particularly useful in calculus for limits and integration.

Process of Redefining Function’s Value

1. Identify the Point: Determine the point(s) where the function needs modification. This could be where the function is undefined or discontinuous.
2. Evaluate the Limit: If the function is approaching a specific value near this point, that value is often used for redefinition.
3. Redefine the Function: Change the function’s definition at the identified point to the evaluated limit or a value that ensures continuity/differentiability.
4. Verify Properties: After redefinition, check that the function now possesses the desired properties, such as continuity or differentiability.

Examples:

Redefining to Remove Discontinuity:

• Original Function: \( f(x) = \frac{x^2 – 1}{x – 1} \).
• At \( x = 1 \), the function is undefined (division by zero).
• However, \( \lim_{x \to 1} f(x) = 2 \).
• Redefined Function: \( f(x) = \begin{cases}
  \frac{x^2 – 1}{x – 1}, & \text{if } x \neq 1 \\
  2, & \text{if } x = 1
  \end{cases} \).

Redefining to Extend Domain:

• Original Function: \( g(x) = \sqrt{x} \) (defined only for \( x \geq 0 \)).
• Redefine for negative \( x \) to extend domain: \( g(x) = \sqrt{|x|} \).