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प्रश्न
Evaluate the following limits: `lim_(x -> 0) [(sqrt(6 + x + x^2) - sqrt(6))/x]`
उत्तर
`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^2)/(x(sqrt(6 + x + x^2) + sqrt(6))`
= `lim_(x -> 0) (x (1 + x))/(x(sqrt(6 + x + x^2) + sqrt(6))`
= `lim_(x -> 0) (1 + x)/(sqrt(6 + x + x^2) + sqrt(6)` ...[∵ x → 0, ∴ x ≠ 0]
= `((1 + 0))/(sqrt(6) + sqrt(6)`
= `1/(2sqrt(6))`
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