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Question
Evaluate the following: `lim_(x -> 2) [(3^(x/2) - 3)/(3^x - 9)]`
Solution
`lim_(x -> 2) [(3^(x/2) - 3)/(3^x - 9)]`
= `lim_(x -> 2)[(3^(x/2) - 3)/((3^(x/2))^2 - (3)^2)]`
= `lim_(x -> 2)(3^(x/2) - 3)/((3^(x/2) - 3)(3^(x/2) + 3)`
= `lim_(x -> 2)1/(3^(x/2) + 3) ...[("As" x -> 2"," x/2 -> 1),(therefore 3^(x/2) -> 3^1 therefore 3^(x/2) ≠ 3),(therefore 3^(x/2) - 3 ≠ 0)]`
= `1/(3^(2/2) + 3)`
= `1/(3^1 + 3)`
= `1/6`
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