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Question
Evaluate the following limit :
`lim_(u -> 1) [(u^4 - 1)/(u^3 - 1)]`
Solution
`lim_(u -> 1) [(u^4 - 1)/(u^3 - 1)]`
= `lim_(u -> 1) ((u^4 - 1^4)/(u - 1))/((u^3 - 1^3)/(u - 1)) ...[(because u -> 1"," therefore u ≠ 1","),(therefore u - 1 ≠ 0)]`
= `(lim_(u -> 1) ((u^4 - 1^4)/(u - 1)))/(lim_(u -> 1)((u^3 - 1^3)/(u - 1))`
= `(4(1)^3)/(3(1)^2) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `4/3`
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