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Question
Evaluate: `lim_(x -> 3) (x^3 + 27)/(x^5 + 243)`
Solution
Given that `lim_(x -> 3) (x^3 + 27)/(x^5 + 243)`
= `lim_(x -> 3) ((x^3 + (3)^3)/(x - 3))/((x^5 + (3)^5)/(x - 3))` ......[Dividing the Nr and Den. by x – 3]
= `(lim_(x -> 3) ((x^3 - (-3)^3)/(x + 3)))/(lim_(x -> 3) ((x^5 - (-3)^2)/(x + 3))` ......`[lim_(x -> a) (f(x))/(g(x)) = (lim_(x -> a) f(x))/(lim_(x -> a) g(x))]`
= `(3(-3)^(3-1))/(5(-3)^(5 - 1))`
= `(3 xx (-3)^2)/(5 xx (-3)^4)`
= `1/(5 xx 3)`
= `1/15`
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