Question

Evaluate the following limit :

`lim_(x -> 2) [(x^3 - 7x + 6)/(x^3 - 7x^2 + 16x - 12)]`

Answer 1

Consider x³ – 7x + 6

2	1  0  -7   6 ↓  2   4  -6
	1  2  -3   0

∴ x³ – 7x + 6 = (x – 2)(x² + 2x – 3)

Consider x³ – 7x² + 16x – 12

2	1  -7   16 -12 ↓   2  -10  12
	1  -5    6     0

∴  x³ – 7x² + 16x – 12 = (x – 2)(x² – 5x + 6)

= (x – 2)(x – 2)(x – 3)

∴ `lim_(x -> 2) (x^3 - 7x + 6)/(x^3 - 7x^2 + 16x - 12)`

= `lim_(x  -> 2) ((x - 2)(x^2 + 2x  - 3))/((x - 2)(x - 2)(x - 3))`

= `lim_(x -> 2) (x^2 + 2x - 3)/((x - 2)(x - 3))`   ...[∵ x → 2, x ≠ 2 ∴ x – 2 ≠ 0]

= `lim_(x -> 2) (x^2 + 2x - 3)` = 2² + 2(2) – 3 = 5 and

`lim_(x -> 2) (x - 2)(x - 3)` = (2 – 2)(2 – 3) = 0

∴  the given limit does not exist.