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Question
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 - x) - 1)/x^2`
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Solution
`lim_(x -> 0) (sqrt(1 - x) - 1)/x^2 = lim_(x -> 0) [(sqrt(1 - x) - 1)/x^2 xx (sqrt(1 - x) + 1)/(sqrt(1 - x) + 1)]`
= `lim_( -> 0) [((1 - x) - 1)/(x^2 (sqrt(1 - x) + 1))]`
= `lim_(x -> 0) [(- x)/(x(sqrt(1 - x) + 1))]`
= `- lim_(x -> 0) [1/(x(sqrt(1 - x) + 1))]`
= `- 1/(0(sqrt(1 - 0) + 1))`
= `- oo`
∴ `lim_(x -> 0) (sqrt(1 - x) - 1)/x^2` does not exist.
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