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