SUMMARY
The limit of the expression \(\lim_{x \to 4}\frac{4 - x^2}{2 - \sqrt{x}}\) is evaluated using algebraic manipulation. By multiplying the numerator and denominator by the conjugate \(2 + \sqrt{x}\), the expression simplifies to \(x(2 - \sqrt{x})\). At \(x = 4\), this limit evaluates to zero, confirming that the limit is not indeterminate. The initial confusion stemmed from an incorrect expansion of the numerator, which should be expressed as \(4 - x^2 = (2 - x)(2 + x)\).
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with algebraic manipulation of expressions
- Knowledge of conjugates in rational expressions
- Ability to identify indeterminate forms
NEXT STEPS
- Study the concept of limits and continuity in calculus
- Learn about indeterminate forms and L'Hôpital's Rule
- Practice algebraic techniques for simplifying rational expressions
- Explore the use of conjugates in limit evaluation
USEFUL FOR
Students studying calculus, particularly those focusing on limits and algebraic manipulation, as well as educators looking for examples of limit evaluation techniques.