Question

The expression x²-4/x³-8 in the simplest form is​

Answer 1

Answer:

Solution:

Using the formula a³ - b³ = (a-b) (a² + ab + b²) where a = x and b = 2,

x³ - 8 = (x)³ - (2)³ = (x - 2) (x² + 2x + 2²)

Using the formula (a² - b²) = (a + b) (a - b) for a = x and b = 2,

x² - 4 = x² - 2² = (x+2) (x-2)

∴ (x³ - 8) ÷ (x² - 4)

= (x - 2) (x² + 2x + 2²)/(x+2) (x-2)

Cancel x-2 from numerator & denominator, & take limit as x approaches 2. We get

Lim {(x³ - 8) ÷ (x² - 4)} as x → 2

= Lim (x² + 2x + 2²)/(x+2) as x → 2

= (2² + 2 x 2 + 2²)/(2+2)

= (4+4+4)/4 = 3x4/4 = 3 (Proved)