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Question
``Evaluate the following limits: `lim_(x -> 4) [(x^2 + x - 20)/(sqrt(3x + 4) - 4)]`
Solution
`lim_(x -> 4) [(x^2 + x - 20)/(sqrt(3x + 4) - 4)]`
= `lim_(x -> 4) [((x + 5)(x - 4))/(sqrt(3x + 4) - 4) xx(sqrt(3x + 4) + 4)/(sqrt(3x + 4) + 4)]`
= `lim_(x -> 4) ((x + 5)(x - 4)(sqrt(3x + 4) + 4))/(3x + 4 - 16)`
= `lim_(x -> 4) ((x + 5)(x - 4)(sqrt(3x + 4) + 4))/(3x - 12)`
= `lim_(x -> 4) ((x + 5)(x - 4)(sqrt(3x + 4) + 4))/(3(x - 4)`
= `lim_(x -> 4) ((x - 5)(sqrt(3x + 4) + 4))/3 ...[(because x ->4"," x ≠ 4),(therefore x - 4 ≠ 0)]`
= `((4 + 5)(sqrt(3(4) + 4) + 4))/3`
= `(9(4 + 4))/3`
= 3(8)
= 24
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