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Question
Evaluate the following limits:
`lim_("h" -> 0) (sqrt(x + "h") - sqrt(x))/"h", x > 0`
Solution
`lim_("h" -> 0) (sqrt(x + "h") - sqrt(x))/"h", x > 0`
`lim_("h" -> 0) (sqrt(x + "h") - sqrt(x))/"h" = lim_(x + "h" -> x) ((x + "h")^(1/2) - x^(1/2))/((x + "h") - x)`
x + h → x
⇒ h → 0
`lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)`
= `1/2(x)^(1/2 - 1)`
= `1/2(x)^(-1/2)`
`lim_("h" -> 0) (sqrt(x + "h") - sqrt(x))/"h" = 1/2 xx 1/(x^(1/2))`
= `1/(2sqrt(x))`
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