The integer n for which ` lim_(x to 0) ((cos x - 1)(cos x -e^(x)))/x^(2) ` is a finite non-zero number, isA. 1B. 2C. 3D. 4

The integer n for which ` lim_(x to 0) ((cos x - 1)(cos x -e^(x)))/x^(

1.	The integer n for which ` lim_(x to 0) ((cos x - 1)(cos x -e^(x)))/x^(2) ` is a finite non-zero number, isA. 1B. 2C. 3D. 4
Answer» Correct Answer - C `underset( x to 0)lim ((cos x - 1)(cos x - e^(x)))/x^(2) ` `underset(x to 0) lim((-2 sin^(2). x/2){(1-x^(2)/(2!) +x^(4)/(4!)-...)-(1+x+x^(2)/(2!) +x^(3)/(3!)+...)})/x^(n)` `underset(x to 0)lim((-2 sin^(2) x/2)(-x-(2x^(2))/(2!) -x^(3)/(3!)-...))/(4(x/2)^(2) x^(n-2))` `underset(x to 0) lim (sin^(2). x/2(1+x+x^(2)/(3!)+...))/(2(x/2)^(2) x^(n-3))` Above limit is finite, if n - 3 = 0, i.e., n = 3.