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Question
Evaluate the following limit :
`lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
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Solution
`lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
= `lim_(x -> 0) [(1 + x/3)/(1 - x/3)]^(1/x) ...[("Divide numerator and"),("denominator 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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\[\lim_{x\to0}\frac{\mathrm{e}^{\tan x}-\mathrm{e}^{x}}{\tan x-x}=\]
