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Question
Evaluate the following limit :
`lim_(x -> 0)[(5x + 3)/(3 - 2x)]^(2/x)`
Solution
`lim_(x -> 0)[(5x + 3)/(3 - 2x)]^(2/x)`
= `lim_(x -> 0) [(3 + 5x)/(3 - 2x)]^(2/x)`
= `lim_(x -> 0) [(1 + (5x)/3)/(1 - (2x)/3)]^(2/x)`
= `lim_(x -> 0)((1 + (5x)/3)^(2/x))/((1 - (2x)/3)^(2/x))`
= `(lim_(x -> 0) (1 + (5x)/3)^(2/x))/(lim_(x -> 0) (1 - (2x)/3)^(2/x))`
= `(lim_(x -> 0)[(1 + (5x)/3)^(3/(5x))]^(10/3))/(lim_(x -> 0)[(1 - (2x)/3)^((-3)/(2x))]^((-4)/3)`
= `([lim_(x -> 0) (1 + (5x)/3)^(3/(5x))]^(10/3))/([lim_(x -> 0) (1 - (2x)/3)^((-3)/(2x))]^((-4)/3)`
= `("e"^(10/3))/("e"^((-4)/3)) ...[(because x -> 0 therefore (5x)/3 -> 0"," (-2x)/3 -> 0),(and lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^(14/3)`.
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