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Question
Evaluate the following limits:
`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)`
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Solution
`lim_(x -> 0) [(sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)] = lim_(x -> 0) [(sqrt(x^2 + 1) - 1) xx ((sqrt(x^2 + 1) + 1))/((sqrt(x^2 + 1) + 1)) xx 1/sqrt(x^2 + 16 - 4) xx (sqrt(x^2 + 16) + 4)/(sqrt(x^2 + 16) + 4)]`
= `lim_(x -> 0) [(x^2 + 1 - 1)/(sqrt(x^2 + 1) + 1) xx (sqrt(x^2 + 16) + 4)/(x^2 + 16 - 16)]`
= `lim_(x -> 0) [x^2/(sqrt(x^2 + 1) + 1) xx (sqrt(x^2 + 16) + 4)/x^2]`
= `lim_(x -> 0) [(sqrt(x^2 + 16) + 4)/(sqrt(x^2 + 1) + 1)]`
= `(sqrt(0^2 + 16) + 4)/(sqrt(0^2 + 1) + 1)`
= `(4 + 4)/(1 + 1)`
`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4) = 8/2` = 4
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