`lim_(xrarr0) (2^(|x|)e^(|x|)-|x|log_(2)2-1)/(xtanx)` is equal to

Options

A. `(1)/(2)(In 2)^2+(1)/(2)(In 2)+1`
B. `(In 2)^2+(1)/(2)+(1)/(2)(In2)+1`
C. `(In 2)^2+(In 2)+(1)/(2)`
D. `(1)/(2)(In 2)^2+(In 2)+(1)/(2)`

Answer

D

Explanation

`lim_(xto0) (2^(|x|)e^(|x|)-|x|log_(2)2-1)/(xtanx)`
`lim_(xto0) (2^(|x|)-|x|log_(2)(2e)-1)/(x^2((tanx)/x))`
`=lim_(xto0) (1+|x|log_e(2e)+(|x|^2)/(2)(log_e(2e))^2+....-|x|log_e(2e)-1)/(x^2((tanx)/x))`
`=lim_(xto0)((|x|^2)/(2)(log_e(2e))^2+....)/(x^2((tanx)/x))=(1)/(2!)(log_e(2e))^2=(1)/(2)(log_e2+1)^2=(1)/(2)(In 2 +1)^2`