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

Redefining a function's value is a mathematical technique often used to address discontinuities, particularly removable discontinuities, or to extend a function's domain. This process involves changing the function's definition at specific points to achieve continuity, differentiability, or other desired properties.

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: $$g(x) = \sqrt{|x|}$$.

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