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Question
Evaluate the following limit :
`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`
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Solution
`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`
= `lim_(x -> 7)[(((x^3 - 343)/(x - 7)))/(((sqrt(x) - sqrt(7))/(x - 7)))] ...[(because x -> 7"," x ≠ 7),(therefore x - 7 ≠ 0)]`
= `(lim_(x -> 7) ((x^3 - 7^3)/(x - 7)))/(lim_(x -> 7) ((x^(1/2) - 7^(1/2))/(x - 7))`
= `(3*(7)^2)/((1)/(2)*(7)^(-1/2)) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `6*(7)^(5/2)`
= `6 xx 7^2 xx sqrt(7)`
= `294sqrt(7)`.
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