limit x to infinity (1+1/6x)^4x+1

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to substitute oo for x in equation under limit such that:

`lim_(x-gtoo) (1+1/6x)^(4x+1) = (1+1/oo)^(oo)`

`lim_(x-gtoo) (1+1/6x)^(4x+1) = (1+0)^(oo) = 1^oo`

Notice that the limit is indeterminate, hence you need to create the special limit `lim_(x-gtoo)(1 + 1/x)^x = e`  such that:

`lim_(x-gtoo) ((1+1/6x)^(6x))^((4x+1)/(6x)) `

Notice that `lim_(x-gtoo) ((1+1/6x)^(6x)) = e` , hence, evaluating `lim_(x-gtoo) ((1+1/6x)^(6x))^((4x+1)/(6x))`  yields `e^(lim_(x-gtoo) ((4x+1)/(6x)))` `lim_(x-gtoo) (4x+1)/(6x)= lim_(x-gtoo) (x(4+1/x))/(6x)` 

Reducing by x yields:

`lim_(x-gtoo) (4+1/x)/6 =lim_(x-gtoo) 4/6 + lim_(x-gtoo) 1/(6x)`

`lim_(x-gtoo) (4+1/x)/6 = 2/3 + 0`

`lim_(x-gtoo) ((1+1/6x)^(6x))^((4x+1)/(6x)) = e^(2/3)`

`lim_(x-gtoo) ((1+1/6x)^(6x))^((4x+1)/(6x)) = root(3)(e^2)`

Hence, evaluating the limit of the function yields `lim_(x-gtoo) ((1+1/6x)^(6x))^((4x+1)/(6x)) = root(3)(e^2).`