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Question
Evaluate the following limits: `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 - 3^3)]`
= `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)^2/((x - 3)(x^2 + 3x + 9))]`
= `lim_(x -> 3)[(x - 3)/(x^2 + 3x + 9)] ...[(because x ->3"," x ≠ 3),(therefore x - 3 ≠ 0)]`
= `(3 - 3)/((3)^2 + 3(3) + 9)`
= `0/27`
= 0
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