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Question
Evaluate the following limits: `lim_(x -> 5)[(x^3 - 125)/(x^2 - 25)]`
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Solution
`lim_(x -> 5)(x^3 - 125)/(x^2 - 25)`
= `lim_(x -> 5) ((x^3 - 125)/(x^2 - 25))/((x^2 - 25)/(x - 5)) ...[("As" x -> 5"," x ≠ 5),(therefore x - 5 ≠ 0),("Divide Numerator and"),("Denominator by" x - 5.)]`
= `lim_(x -> 5)(((x^3 - 5^3)/(x - 5)))/(((x^2 - 5^2)/(x - 5))`
= `(3(5)^2)/(2(5)^1) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `15/2`
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