lim

Apuntes

• Subtopic: Limit Calculation Evaluate the limit as \( x \) approaches 3: \[ \lim_{x \to 3} \frac{x^2 + 2x - 15}{x^3 - 6x^2 + 11x - 6} \] First, factor the numerator and denominator to simplify the expression. Numerator: \( x^2 + 2x - 15 \) Factor: \[ x^2 + 2x - 15 = (x + 5)(x - 3) \] Denominator: \( x^3 - 6x^2 + 11x - 6 \) Factor: \[ x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) \] Now, rewrite the limit: \[ \lim_{x \to 3} \frac{(x + 5)(x - 3)}{(x - 1)(x - 2)(x - 3)} \] Cancel the common factor \( (x - 3) \): \[ \lim_{x \to 3} \frac{(x + 5)}{(x - 1)(x - 2)} \] Substitute \( x = 3 \): \[ \frac{3 + 5}{(3 - 1)(3 - 2)} = \frac{8}{(2)(1)} = \frac{8}{2} = 4 \] **Final answer:** \[ \boxed{4} \]