The value of `lim_(x rarr 0) (1-cos2x)/(e^(x^(2))-e^(x)+x)` is

Options

A. 0
B. 2
C. 4
D. 8

Answer

C

Explanation

Let `L=underset(xrarr0)(lim)(e^(x^(2))-e^(x)+x)/(1-cos2x)`
`=underset(xrarr0)(lim)(e^(x^(2))-1+x-e^(x)+1)/(2(sin^(2)x)/(x^(2)).x^(2))`
`=(1)/(2)[underset(xrarr0)(lim)(e^(x^(2))-1)/(x^(2))+underset(xrarr0)(lim)(x-e^(x)+1)/(x^(2))]`
`=(1)/(2)[1-underset(xrarr0)(lim)(e^(x)-1)/(2x)]" (using L' Hospital Rule)"`
`=(1)/(2)[1-(1)/(2)]`
`=(1)/(4)`
`rArr" "underset(xrarr0)(lim)(1-cos2x)/(e^(x^(2))-e^(x)+x)=4`