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Question
Evaluate the Following limit :
`lim_(x -> 0)[3/(xsqrt(9 - x)) - 1/x]`
Solution
`lim_(x -> 0)[3/(xsqrt(9 - x)) - 1/x]`
= `lim_(x -> 0)[(3 - sqrt(9 - x))/(xsqrt(9 - x))]`
= `lim_(x -> 0)[(3 - sqrt(9 - x))/(xsqrt(9 - x)) xx (3 + sqrt(9 - x))/(3 + sqrt(9 - x))]`
= `lim_(x -> 0)[(9 - (9 - x))/(xsqrt(9 - x)(3 + sqrt(9 - x)))]`
= `lim_(x -> 0) [x/(xsqrt(9 - x)(3 + sqrt(9 - x)))]`
= `lim_(x -> 0) [1/(sqrt(9 - x)(3 + sqrt(9 - x)))]` ...[∵ x → 0, ∴ x ≠ 0]
= `1/(sqrt(9 - 0)(3 + sqrt(9 - 0))`
= `1/(3(3 + 3))`
= `1/18`
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