Question

What is the integral nalue of `n` for which `lim_(x->0)[cos^2x-cosx-e^xcosx+e^x-x^3/3]/x^n` is a finite non-zero number?

Answer 1

`n=3`

Explanation:

Calling

`f(x) = cos^2x-cosx-e^xcosx+e^x-x^3/3`

Near `x = 0` we have

`f(x) = f(0)+f'(0)x+f''(0)x^2/(2!)+f^((3))(0)x^3/(3!)+O(x^4)`

but `f(0)=f'(0) = f''(0)=0` then

`f(x) = x^3/6 + O(x^4)` then for small `abs x` we have

`lim_(x->0)f(x)/x^n = lim_(x->0) (x^3/6 + O(x^4))/x^n` then for `n = 3` we have

`lim_(x->0)f(x)/x^3 =1/6`