The value of `lim_(xrarr0) ((1+2x)/(1+3x))^((1)/(x^(2))).e^((1)/(e^(x)))` is

Options

A. `e((5)/(2))`
B. `e^(2)`
C. `e^(-2)`
D. 1

Answer

A

Explanation

`underset(xrarr0)(lim)e^((1)/(x^(2))log(1+2x)-log(1+3x)+(1)/(x))`
`=e^(underset(xrarr0)(lim)(log(1+2x)-log(1+3x)+x)/(x^(2)))`
`=e^(underset(xrarr0)(lim)(2x-((2x)^(2))/(2)-3x+((3x)^(2))/(2)+x)/(x^(2)))" (using expansion)"`
`=e^(underset(xrarr0)(lim)((5)/(2)x^(2))/(x^(2)))`
`=e^((5)/(2))`