Evaluate the limit using L'Hospital's rule if necessary lim as x approaches infinity of ((12x)/(12x+9))^(9x)    

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should add and subtract 9 to numerator such that:

`lim_(x-gtoo)((12x + 9)/(12x + 9) - 9/(12x + 9))^(9x)`

`lim_(x-gtoo)(1+ (-9/(12x + 9)))^(9x) = 1^oo`

You may use the special limit `lim_(x-gtoo)(1 + 1/x)^x = e`  such that:

`lim_(x-gtoo)((1+ (-9/(12x + 9)))^((12x+9)/(-9)))^((-81x)/(12x+9)) = e^ lim_(x-gtoo)((-81x)/(12x+9))`

`lim_(x-gtoo)((-81x)/(12x+9)) = -81/12 = -27/4`

Hence, evaluating the limit of the function yields `lim_(x-gtoo)((12x)/(12x+9))^(9x) = e^(-27/4).`

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