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Question
Evaluate the following limits: `lim_(x -> 1) [(x^2 + xsqrt(x) - 2)/(x - 1)]`
Solution
`lim_(x -> 1) [(x^2 + xsqrt(x) - 2)/(x - 1)]`
= `lim_(x -> 1) [((x^2 - 1) + (xsqrt(x) - 1))/(x - 1)]`
= `lim_(x -> 1)[(x^2 - 1)/(x - 1) + (x^(3/2) - 1)/(x - 1)] ...[because xsqrt(x) = x^1*x^(1/2) = x^(1 + 1/2) = x^(3/2)]`
= `lim_(x -> 1) ((x^2 - 1^2)/(x - 1)) + lim_(x -> 1)((x^(3/2) - 1^(3/2))/(x - 1))`
= `2(1)1 3/2(1)^(1/2) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `2 + 3/2`
= `7/2`
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