Evaluate the following limit : lim(x→2) (x^3 + 3x^2 - 9x - 2)/(x^3 - x - 6)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 2}\cfrac{\text x^3+3\text x^2-9\text x-2}{\text x^3 - \text x-6} \)

lim(x→2) (x³ + 3x² - 9x - 2)/(x³ - x - 6)

Answer 1

Since the form is indeterminant

\(=\cfrac00\)

Method 1: factorization

\(\lim\limits_{\text x \to 2}\cfrac{\text x^3+3\text x^2-9\text x-2}{\text x^3 - \text x-6} \)

By long division method

Since a³ - b³ = (a - b) (a² + b² + ab) & a² - b² = (a + b) (a - b)

Method2: By L hospital rule:

Differentiating numerator and denominator separately: