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Question
Evaluate the following limit :
`lim_(x -> 1) [(x^4 - 3x^2 + 2)/(x^3 - 5x^2 + 3x + 1)]`
Solution
`lim_(x -> 1) [(x^4 - 3x^2 + 2)/(x^3 - 5x^2 + 3x + 1)]`
To find the factor of numerator and denominator by synthetic division
Consider, numerator = x4 + 0x3 – 3x2 + 0x + 2
| 1 |
1 0 -3 0 2 1 1 -2 -2 |
| 1 1 -2 -2 0 |
∴ numerator = (x – 1) (x3 + x2 – 2x – 2)
Now, denominator = x3 – 5x2 + 3x + 1
| 1 |
1 -5 3 1 1 -4 -1 |
| 1 -4 -1 0 |
∴ denominator = (x – 1) (x2 – 4x – 1)
∴ `lim_(x -> 1) (x^4 - 3x^2 + 2)/(x^3 - 5x^2 + 3x + 1)`
= `lim_(x -> 1) ((x - 1)(x^3 + x^2 - 2x - 2))/((x - 1)(x^2 - 4x - 1))`
= `lim_(x -> 1) (x^3 + x^2 - 2x - 2)/(x^2 - 4x - 1)` ...[∵ x → 1, ∴ x ≠ 1, ∴ x − 1 ≠ 0)]
= `(lim_(x -> 1) (x^3 + x^2 - 2x - 2))/(lim_(x -> 1) (x^2 - 4x - 1))`
= `(1^3 + 1^2 - 2(1) - 2)/(1^2 - 4(1) - 1)`
= `(1 + 1 - 2 - 2)/(1 - 4 - 1)`
=`(-2)/(-4)`
= `1/2`
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