Question

The integer n, for which  lim_(x-->0) (cosx-1)(cosx-e^x)/ xⁿ  is a finite non zero number is

Answer 1

As we know that  lim_(x-->0)(1 - cos x)/x² = 1/2

and also  lim_(x-->0)(e^x - 1)/x = 1

So convert the denominator accordingly and solve the same.

Answer 2

for this limit to be finite at x = 0, x³ should be eliminated  hence xⁿ = x³ , n = 3

Hence option (c) is correct.