Question

If the value of `lim_(x→0)(2 - cosxsqrt(cos2x))^(((x + 2)/x^2))` is equal to e^a, then a is equal to ______.

Options

• 0

• 1

• 2

• 3

Answer 1

If the value of `lim_(x→0)(2 - cosxsqrt(cos2x))^(((x + 2)/x^2))` is equal to e^a, then a is equal to 3.

Explanation:

`lim_(x→0)(2 - cosxsqrt(cos2x))^(((x + 2)/x^2))`

From indeterminate form 1^∞

= `e^(lim_(x →0)((1 - cosxsqrt(cos2x))/x^2)(x + 2)`

= `e^(lim_(x →0)((1 - cos^2x.cos2x)/(x^2(1 + cosxsqrt(cos2x))))(x + 2)`

= `e^(lim_(x →0)((1 - (1 sin^2x)(1 - 2sin^2x))(x + 2))/(x^2(1 + cosxsqrt(cos2x))`

= `e^(lim_(x →0)((3sin^2x - 2sin^4x)(x + 2))/(x^2(1 + cosxsqrt(cos2x))`

= = `e^(lim_(x →0)((sin^2x)/x^2)((3 - 2sin^2x)(x + 2))/(1 + cosxsqrt(cos2x))`

= `e^((3 xx 2)/2)` = e³

∴ a = 3