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Question
Select the correct answer from the given alternatives.
`lim_(x -> 0) ((3 + 5x)/(3 - 4x))^(1/x)` =
Options
e3
e6
e9
e-3
Solution
`lim_(x -> 0) ((3 + 5x)/(3 - 4x))^(1/x)` = e3
Explanation:
`lim_(x -> 0) ((3 + 5x)/(3 - 4x))^(1/x)`
= `lim_(x -> 0)((1 + (5x)/3)/(1 - (4x)/3))^(1/x)` ...[Divide numberator and denominator by 3]
= `(lim_(x -> 0)[(1 + (5x)/3)^(3/(5x))]^(5/3))/(lim_(x -> 0)[(1 - (4x)/3)^((-3)/(4x))]^((-4)/3))`
= `e^(5/3)/e^((-4)/3)`
= e3
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