# 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

**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.

**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.**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

**Identify the Point**: Determine the point(s) where the function needs modification. This could be where the function is undefined or discontinuous.**Evaluate the Limit**: If the function is approaching a specific value near this point, that value is often used for redefinition.**Redefine the Function**: Change the function’s definition at the identified point to the evaluated limit or a value that ensures continuity/differentiability.**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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