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Question
Evaluate the following Limits: `lim_(x -> "a") ((x + 2)^(5/3) - ("a" + 2)^(5/3))/(x - "a")`
Solution
`lim_(x -> "a") ((x + 2)^(5/3) - ("a" + 2)^(5/3))/(x - "a")`
Put x + 2 = y and a + 2 = b
As x → a, x + 2 → a + 2
i.e. y → b.
∴ `lim_(x -> "a") ((x + 2)^(5/3) - ("a" + 2)^(5/3))/(x - "a")`
= `lim_(x -> "a") (y^(5/3) - "b"^(5/3))/((y - 2) - ("b" - 2)`
= `lim_(y -> "b") (y^(5/3) - "b"^(5/3))/(y - "b")`
= `5/3"b"^(2/3) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `5/3("a" + 2)^(2/3)` ...[∵ b = a + 2]
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