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Question
Evaluate the following: `lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
Solution
`lim_(x -> 0) ((3 + x)/(3 - x))^(1/x)`
= `lim_(x -> 0) ((1 + x/3)/(1 - x/3))^(1/x) ...[("Divide numerator and"),("denominato by" 3)]`
= `lim_(x -> 0)((1 + x/3)^(1/x))/((1 - x/3)^(1/x)`
= `lim_(x -> 0) ((1 + x/3)^(3/x xx 1/3))/((1 - x/3)^((-3)/x xx 1/(-3))`
= `(lim_(x -> 0)[(1 + x/3)^(3/x)]^(1/3))/(lim_(x -> 0)[(1 - x/3)^((-3)/x)]^(-1/3)`
= `("e"^(1/3))/("e"(-1)/(3)) ...[(because x -> 0"," x/3 -> 0"," (-x)/3 -> 0 and),(lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^(1/3 + 1/3)`
= `"e"^(2/3)`
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