The value of `lim_(xtooo) ((20^(x)-1)/(19(5^(x))))^(1//x)` is ________.

Options

A. 3
B. 1
C. `(2)/(3)`
D. `(3)/(2)`

Answer

`(4)`

Explanation

We have,
`y=underset(xtooo)lim((20^(x)-1)/(19(5^(x))))^((1)/(x))" "(oo^(0)" form")`
`:." "log_(e)y=log_(e)underset(xtooo)lim((20^(x)-1)/(19(5^(x))))^((1)/(x))`
`=underset(xtooo)limlog_(e)((20^(x)-1)/(19(5^(x))))^((1)/(x))`
`=underset(xtooo)lim(1)/(x)log_(e)((20^(x)-1)/(19(5^(x))))`
`=underset(xtooo)lim(log_(e)(20^(x)-1)-log_(e)19-log_(e)5^(x))/(x)`
`=underset(xtooo)lim((20^(x)xxlog_(e)20)/(20^(x)-1)-0-(5^(x)log_(e)5)/(5^(x)))/(1)`
`=underset(xtooo)lim(((log_(e)20)/(1))/(1-(1)/(20^(x)))-log_(e)5)`
`=log_(e)20-log_(e)5=log_(e)4`