The integer n for which `lim_(x rarr 0) ((cos x-1) ( cos x - e^x))/x^n` is a finite non-zero number is :

Explanation

Here, we will use the expansion,

`cosx = 1-x^2/(2!)+x^4/(4!)-x^6/(6!)+...`


`lim_(x->0) ((cosx-1)(cosx-e^x))/x^n`

`=lim_(x->0) ((1-x^2/2+...-1)(1-x^2/2+...-(1+x+x^2/2+...)))/x^n`

Here, we will not use higher `x` terms as `x->0`

So, given expression becomes,

`=lim_(x->0) ((1+x^2/2-1)(1-x^2/2-1-x-x^2/2))/x^n`

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