Evaluating Absolute Value Function Limits Graphically: Calculus Example

Understanding Absolute Value Functions

• 💡 The absolute value function, like |x - 3|, can be represented as a piecewise function.
• 🎯 For |x - 3|, it simplifies to x - 3 when x > 3 and -(x - 3) when x < 3.

Graphing the Absolute Value Function

• 🚀 The graph of |x - 3| / (x - 3) is shifted three units to the right from the parent function |x| / x.
• 📈 When x > 3, the function's value is positive 1.
• 📉 When x < 3, the function's value is negative 1.
• ⚠️ A jump discontinuity occurs at x = 3.

Evaluating the Limit

• 🔍 The limit as x approaches 3 from the left side is negative 1.
• 🔍 The limit as x approaches 3 from the right side is positive 1.
• ❌ Because the left-sided and right-sided limits do not match, the overall limit does not exist at x = 3.