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Question
Evaluate the following :
`lim_(x -> 1) [(sqrt(x) - 1)/logx]`
Solution
`lim_(x -> 1) (sqrt(x) - 1)/logx`
put x – 1 = h
∴ x = 1 + h
As x → 1, h → 0
∴ Required limit
= `lim_("h" -> 0) (sqrt(1 + "h") - 1)/(log(1 + "h"))`
= `lim_("h" -> 0) (sqrt(1 + "h") - 1)/(log (1 + "h")) xx (sqrt(1 + "h") + 1)/(sqrt(1 + "h") + 1)`
= `lim_("h" -> 0) ((1 + "h") - 1)/(log(1 + "h")) xx 1/(sqrt(1 + "h") + 1)`
= `lim_("h" -> 0) "h"/(log(1 + "h")) xx 1/(sqrt(1 + "h") + 1)`
= `lim_("h" -> 0) (1)/((log(1 + "h"))/("h")) xx 1/(sqrt(1 + "h") + 1) ...[("Divide Numerator and"),("Denominator by h"),("As" "h" -> 0"," "h" ≠ 0)]`
= `lim_("h" -> 0) (1)/((log(1 + "h"))/("h")) xx 1/(lim_("h" -> 0)(sqrt(1 + "h") + 1)`
= `1/1 xx 1/(sqrt(1 + 0) + 1)`
= `1/2`
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