Knowledge of expansion of many functions expands your horizon of understanding calculus

I came across a question (IIT - 2002), which read as:

What is the integer n for which limit x tends to 0 for [{cos x - 1}{cos x – e^x}/xⁿ] is a finite non-zero number? The options were: 1 or 2 or 3 or 4.
Without expansion, it is rather difficult to solve it. Even application of L'Hospital's Rule will not be a providential help. But if we expand the exponential and trigonometric functions, the question is easily solvable.