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Question
Evaluate the following limit:
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
Solution
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
= `lim_(x -> 5)(((x^3 - 5^3)/(x - 5)))/(((x^5 - 5^5)/(x - 5))) ...[(because x -> 5"," therefore x ≠ 5","),(therefore x - 5 ≠ 0)]`
= `(lim_(x -> 5) (x^3 - 5^3)/(x - 5))/(lim_(x -> 5)(x^5 - 5^5)/(x - 5)`
= `(3(5)^2)/(5(5)^4) ...[ because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "n"*"a"^("n" - 1)]`
= `3/(5)^3`
= `3/125`
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