Find the value of `alpha` so that `("lim")_(xvec0)1/(x^2)(e^(alphax)-e^x-x)=3/2`

Options

A. 1
B. 0
C. 4
D. 2

Answer

D

Explanation

`lim_(xto0) ((e^(ax)-e^x-x))/(x^2)=(3)/(2)`
`rArr lim_(xto0) ({1+alphax+((alphax)^2)/(2!)+......}-{1+x+(x^2)/(2!)+......}-x)/(x^2)=(3)/(2)`
`lim_(xto0) ((alpha-2)x+((alpha^2-1))/(2!)x^2+(x^3)/(3!)(alpha^(3)-1)+..)/(x^2)=(3)/(2)`
`rArr alpha-2=0rArr alpha=2`