# Evaluate the following limit : limx→3[1x-3-9xx3-27] - Mathematics and Statistics

Sum

Evaluate the following limit :

lim_(x -> 3) [1/(x - 3) - (9x)/(x^3 - 27)]

#### Solution

lim_(x -> 3) [1/(x - 3) - (9x)/(x^3 - 27)]

= lim_(x -> 3)  [1/(x - 3) - (9x)/((x - 3)(x^2 + 3x + 9))]

= lim_(x -> 3) [(x^2 + 3x + 9 - 9x)/((x - 3)(x^2 + 3x + 9))]

= lim_(x -> 3) [(x^2 - 6x + 9)/((x - 3)(x^2 + 3x + 9))]

= lim_(x -> 3) ((x - 3)(x - 3))/((x - 3)(x^2 + 3x + 9))

= lim_(x -> 3)  (x - 3)/(x^2 + 3x + 9)   ....[(because x -> 3","  x ≠ 3),(therefore x - 3 ≠ 0)]

= (lim_(x -> 3) (x - 3))/(lim_(x -> 3) (x^2 + 3x + 9))

= (3 - 3)/(3^2 + 3 xx 3 + 9)

= 0

Concept: Factorization Method
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