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Question
Evaluate: `lim_(x -> 2) (x^2 - 4)/(sqrt(3x - 2) - sqrt(x + 2))`
Solution
Given that `lim_(x -> 2) (x^2 - 4)/(sqrt(3x - 2) - sqrt(x + 2))`
Rationalizing the denominator, we get
= `lim_(x -> 2) ((x - 2)(x + 2) [sqrt(3x - 2) + sqrt(x + 2)])/([sqrt(3x - 2) - sqrt(x + 2)][sqrt(3x - 2) + sqrt(x + 2)])`
= `lim_(x -> 2) ((x - 2)(x + 2)[sqrt(3x - 2) + sqrt(x + 2)])/(3x - 2 - x - 2)`
= `lim_(x -> 2) ((x - 2)(x + 2)[sqrt((3x - 2)) + sqrt(x + 2)])/(2x - 4)`
= `lim_(x -> 2) ((x - 2)(x + 2) [sqrt((3x - 2)) + sqrt(x + 2)])/(2(x - 2))`
= `lim_(x -> 2) ((x + 2)[sqrt(3x - 2) + sqrt(x + 2)])/2`
Taking limits, we have
`= ((2 + 2)[sqrt(6 - 2) + sqrt(2 + 2)])/2`
= `(4[2 + 2])/2`
= `(4 xx 4)/2`
= 8
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