If `lim_(xto-2^(-)) (ae^(1//|x+2|)-1)/(2-e^(1//|x+2|))=lim_(xto-2^(+)) sin((x^(4)-16)/(x^(5)+32)),` then a is

Options

A. sin(3/5)
B. sin(2/5)
C. 0
D. none of these

Answer

C

Explanation

`underset(xto-2^(-))lim(ae^(1//|x+2|)-1)/(2-e^(1//|x+2|))=underset(xto2^(-))lim(a-e^(-1//|x+2|))/(2e^(-1//|x+2|)-1)=-a`
`=underset(xto-2^(-))limsin((x^(4)-16)/(x^(5)+32))=underset(xto2^(-))limsin(((x^(4)-(-2)^(4))/(x-(-2)))/((x^(5)-(-2)^(5))/(x-(-2))))`
`=sin(-(2)/(5))`
or `a="sin"2/5`