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Question
Evaluate the following limit:
`lim_(x -> 0)[(sqrt(6 + x + x^2) - sqrt(6))/x]`
Solution
`lim_(x -> 0)(sqrt(6 + x + x^2) - sqrt(6))/x`
= `lim_(x -> 0) (sqrt(6 + x + x^2) - sqrt(6))/x xx (sqrt(6 + x + x^2) + sqrt(6))/(sqrt(6 + x + x^2) + sqrt(6))`
= `lim_(x -> 0) (6 + x + x^2 - 6)/(x[sqrt(6 + x + x^2) + sqrt(6)]`
= `lim_(x -> 0) (x(x + 1))/(x[sqrt(6 + x + x^2) + sqrt(6)]`
= `lim_(x -> 0) (x + 1)/(sqrt(6 + x + x^2) + sqrt(6))` ...[∵ x → 0 ∴ x ≠ 0]
= `(lim_(x -> 0) (x + 1))/(lim_(x -> 0)[sqrt(6 + x + x^2) + sqrt(6)]`
= `(0 + 1)/(sqrt(6 + 0 + 0) + sqrt(6))`
= `1/(2sqrt(6))`
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