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