Question

If `lim_(x→0) (int_0^x((cos2t - 1)(cost - e^(-t^2))t^-n)dr)/(cosx - 1)` is a finite non-zero number, Then the integer value for n is ______.

Options

• 0.00

• 1.00

• 2.00

• 3.00

Answer 1

If `lim_(x→0) (int_0^x((cos2t - 1)(cost - e^(-t^2))t^-n)dr)/(cosx - 1)` is a finite non-zero number, Then the integer value for n is 3.00.

Explanation:

`lim_(x→0)((cos2x - 1)(cosx - e^(-x^2))x^-n)/(-sinx)`  ...(Applying L’ Hopital’s rule)

= `lim_(x→0)(2sin^2x(cosx - e^(-x^2)))/(x^n.sinx)`

= `lim_(x→0)sinx/x((cosx - e^(-x^2)))/(x^(n - 1)`

= `lim_(x→0) sinx/x(((cosx - 1))/x^(n - 1) - ((e^(-x^2 - 1)))/x^(n - 1))`

= `(cosx - 1)/x^(n - 1) → (-1)/2` if n = 3, `(e^(-x^2) - 1)/x^(n - 1) → -1` if n = 3

Hence required value for n = 3