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Question
Evaluate: `lim_(h -> 0) (sqrt(x + h) - sqrt(x))/h`
Solution
Given that `lim_(h -> 0) (sqrt(x + h) - sqrt(x))/h`
= `lim_(h -> 0) (sqrt(x + h) - sqrt(x))/(h[sqrt(x + h) + sqrt(x)]) xx sqrt(x + h) + sqrt(x)` ....[Rationalizing the denominator]
= `lim_(h -> 0) (x + h - x)/(h[sqrt(x + h) + sqrt(x)]`
= `lim_(h -> 0) h/(h[sqrt(x + h) + sqrt(x)]`
= `lim_(h -> 0) 1/(sqrt(x + h) + sqrt(x))`
Taking the limits, we have
`1/(sqrt(x) + sqrt(x)) = 1/(2sqrt(x))`
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