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Question
Evaluate `lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x))`
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Solution
We have `lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x))`
= `lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x)) xx (sqrt(a + 2x) + sqrt(3x))/(sqrt(a + 2x) + sqrt(3x))`
= `lim_(x -> a) (a + 2x - 3x)/((sqrt(3a + x) - 2sqrt(x))(sqrt(a + 2x) + sqrt(3x))`
= `lim_(x -> a) (((a - x))(sqrt(3a + x) + 2sqrt(x)))/((sqrt(a + 2x) + sqrt(3x))(sqrt(3a + x) - 2sqrt(x))(sqrt(3a + x) + 2sqrt(x))`
= `lim_(x -> a) ((a - x) sqrt(3a + x) + 2sqrt(x))/((sqrt(a + 2x) + sqrt(3x))(3a + x - 4x))`
= `(4sqrt(a))/(3 xx 2sqrt(3a))`
= `2/(3sqrt(3))`
= `(2sqrt(3))/9`.
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