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Question
Evaluate the following limit :
`lim_(y -> 1)[(2y - 2)/(root(3)(7 + y) - 2)]`
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Solution
`lim_(y -> 1)[(2y - 2)/(root(3)(7 + y )- 2)]`
= `lim_(y -> 1) [(2y - 2)/((7 + y)^(1/3) - 8^(1/3))] xx [((7 + y)^(2/3) + (7 + y)^(1/3) 8^(1/3) + 8^(2/3))/((7 + y)^(2/3) + (7 + y)^(1/3) 8^(1/3) + 8^(2/3))]`
= `lim_(y -> 1) ((2y - 2)[(7 + y)^(2/3) + (7 + y)^(1/3) 8^(1/3) + 8^(2/3)])/((7 + y) - 8) ...[because "a" - "b" = ("a"^(1/3) - "b"^(1/3)) ("a"^(2/3) + "a"^(1/3)"b"^(1/3) + "b"^(2/3))]`
= `lim_(y -> 1) (2(y - 1)[(7 + y)^(2/3) + (7 + y)^(1/3) 8^(1/3) + 8^(2/3)])/(y - 1)`
= `lim_(y -> 1) 2[(7 + y)^(2/3) + (7 + y)^(1/3) 8^(1/3) + 8^(2/3)] ...[(because y -> 1"," y ≠ 1","),(therefore y - 1 ≠ 0)]`
= `2[(7 + 1)^(2/3) + (7 + 1)^(1/3) 8^(1/3) + 8^(2/3)]`
= = 2[4 + 4 + 4] ...`[∵ 8^(1/3) = 2 and 8^(2/3) = 4]`
= 24
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