Evaluating Limits of Absolute Value Expressions with Piecewise Functions

Evaluating Limits of Absolute Value Expressions

• 🎯 The problem involves finding the limit of an expression with an absolute value as x approaches a specific number.
• 💡 The first step is to define the piecewise function for the absolute value expression.

Left-Hand Limit Evaluation

• ⬅️ For the limit as x approaches 2 from the left, numbers are less than 2, so the absolute value of (x - 2) is replaced with -(x - 2).
• 🧩 The numerator, x² - 4, is factored as a difference of squares: (x + 2)(x - 2).
• ✂️ The (x - 2) terms in the numerator and denominator cancel out.
• ➕ Direct substitution of x = 2 into the remaining expression -(x + 2) yields -4.

Right-Hand Limit Evaluation

• ➡️ For the limit as x approaches 2 from the right, numbers are greater than 2, so the absolute value of (x - 2) is replaced with (x - 2).
• ✂️ Similar to the left-hand limit, the (x - 2) terms cancel out.
• ➕ Direct substitution of x = 2 into the remaining expression (x + 2) yields 4.

Conclusion on Limit Existence

• ⚠️ Since the left-hand limit (-4) and the right-hand limit (4) are not equal, the overall limit does not exist.