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Question
Evaluate the following limit :
`lim_(x -> 0)[(4x + 1)/(1 - 4x)]^(1/x)`
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Solution
`lim_(x -> 0)[(4x + 1)/(1 - 4x)]^(1/x)`
= `lim_(x -> 0) [(1 + 4x)/(1 - 4x)]^(1/x)`
= `lim_(x -> 0) (1 + 4x)^(1/x)/(1 - 4x)^(1/x)`
= `(lim_(x -> 0) (1 + 4x)^(1/x))/(lim_(x -> 0) (1 - 4x)^(1/x))`
= `(lim_(x -> 0) [(1 + 4x)^(1/(4x))]^4)/(lim_(x -> 0)[(1 - 4x)^(-1/(4x))]^-4`
= `([lim_(x -> 0) (1 + 4x)^(1/(4x))]^4)/([lim_(x -> 0) (1 - 4x)^(-1/(4x))]^-4`
= `"e"^4/"e"^(-4) ...[because x -> 0"," ± 4x -> 0 and lim_(x -> 0) (1 + x)^x = "e"]`
= e8.
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