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Question
Evaluate the following limit :
`lim_(x -> 3) [1/(x - 3) - (9x)/(x^3 - 27)]`
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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
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