Find the integral value of `n`
for which
`("lim")_(xvec0)(cos^2x-cosx-e^xcosx+e^x-(x^3)/2)/(x^n)`
is a finite nonzero number
Text Solution
Verified by Experts
Given that `underset(xto0)lim (cos^(2)x-cosx-e^(x)cosx+e^(x)-(x^(3))/(2))/(x^(n))` `=underset(xto0)lim((cosx-1)(cosx-e^(x))-(x^(3))/(2))/(x^(n))` `=underset(xto0)lim((1-(x^(2))/(2!)+(x^(4))/(4!)-(x^(6))/(6!)+...-1)[(1-(x^(2))/(2!)+(x^(4))/(4!)-...)-(1+x+(x^(2))/(2!)+(x^(3))/(3!)...)]-(x^(3))/(2))/(x^(n))` `=underset(xto0)lim((-(x^(2))/(2!)+(x^(4))/(4!)-(x^(6))/(6!)+...)[(-x-x^(2)-(x^(3))/(3!)-(x^(5))/(5!)-...)]-(x^(3))/(2))/(x^(n))` `=underset(xto0)lim(((x^(3))/(2)+(x^(4))/(2)+(x^(5))/(12)-(x^(5))/(24
)+...)-(x^(3))/(2))/(x^(n))` =nonzero if n = 4
