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Question
Evaluate the following limit :
`lim_(x -> 0) [(5 + 7x)/(5 - 3x)]^(1/(3x))`
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Solution
`lim_(x -> 0) [(5 + 7x)/(5 - 3x)]^(1/(3x))`
= `lim_(x -> 0)[(1 + 7/x x)/(1 - 3/5 x)]^(1/(3x)) ...[("Divide numerator and"),("denominator by 5")]`
= `lim_(x -> 0) (1 + (7x)/5)^(1/(3x))/(1 - (3x)/5)^(1/(3x))`
= `(lim_(x -> 0)[(1 + (7x)/5)^(5/(7x))]^(7/5 xx 1/3))/(lim_(x -> 0) [(1 - (3x)/5)^((-5)/(3x))]^((-3)/5 xx 1/3)`
= `"e"^(7/15)/"e"^((-3)/(15)) ...[(because x -> 0"," (7x)/5 -> 0"," (-3x)/5 -> 0 and),(lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^(7/15 + 3/15)`
= `"e"^(10/15)`
= `"e"^(2/3)`
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