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Question
Evaluate the following limits:
`lim_(x - 0) (sqrt(1 + x^2) - 1)/x`
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Solution
`lim_(x - 0) (sqrt(1 + x^2) - 1)/x = lim_(x -> 0) [(sqrt(1 + x^2) - 1)/x xx (sqrt(1 + x^2) + 1)/(sqrt(1 + x^2) + 1)]`
= `lim_(x -> 0) [((1 + x^2) - 1)/(x(sqrt(1 + x^2) + 1))]`
= `lim_(x -> 0) [x^2/(x(sqrt(1 + x^2) + 1))]`
= `lim_(x -> 0) [x/(sqrt(1 + x^2) + 1)]`
= `0/(sqrt(1 + 0^2) + 1)`
`lim_(x -> 0) (sqrt(1 + x^2) - 1)/x = 0/2` = 0
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