Question

Evaluate the following limit:

`lim_(x -> 1) [(x^4 - 3x^2 + 2)/(x^3 - 5x^2 + 3x + 1)]`

Answer 1

`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`