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Question
Evaluate: `lim_(x -> a) ((2 + x)^(5/2) - (a + 2)^(5/2))/(x - a)`
Solution
Given that `lim_(x -> a) ((2 + x)^(5/2) - (a + 2)^(5/2))/(x - a)`
= `lim_(x -> a) ((2 + x)^(5/2) - (a + 2)^(5/2))/((2 + x) - (a + 2))`
= `lim_(2 + x -> a + 2) ((2 + x)^(5/2) - (a + 2)^(5/2))/((2 + x) - (a + 2))`
= `5/2 (a + 2)^(5/(2 - 1))`
= `5/2(a + 2)^(3/2)` ....`[because lim_(x -> a) (x^n - a^n)/(x - a) = n * a^(n - 1)]`
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