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Question
Evaluate the following limit :
`lim_(x -> 2) [(x^3 - 7x + 6)/(x^3 - 7x^2 + 16x - 12)]`
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Solution
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.
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