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Question
Evaluate the following limits: `lim_(y -> 1) [(2y - 2)/(root(3)(7 + y) - 2)]`
Solution
`lim_(y -> 1) (2y - 2)/(root(3)(7 + y) - 2)`
= `lim_(y -> 1) (2(y - 1))/((7 + y)^(1/3) - 8^(1/3)) ...[because 2 = (2^3)^(1/3) = 8^(1/3)]`
= `lim_(y -> 1) 2/(((y + 7)^(1/3) - 8^(1/3))/(y - 1)`
= `(lim_(y -> 1) 2)/(lim_(y -> 1) ((y + 7)^(1/3) - 8^(1/3))/((y + 7) - 8)`
Let y + 7 = x
As y → 1, x → 8
= `2/(lim_(x -> 8) (x^(1/3) - 8^(1/3))/(x - 8))`
= `2/(1/3(8)^((-2)/2)) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `2(3)*(8)^(2/3)`
= `6(2^3)^(2/3)`
= 6 x (2)2
= 24
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