Question

If f(x) = `{{:((sin^3(sqrt(3)).log(1  +  3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3))  -  1)x)",", x ≠ 0),(                         a",", x = 0):}`

is continuous in [0, 1] then a is equal to ______.

Options

• 0

• `3/5`

• 2

• `5/3`

Answer 1

If f(x) = `{{:((sin^3(sqrt(3)).log(1  +  3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3))  -  1)x)",", x ≠ 0),(                         a",", x = 0):}`

is continuous in [0, 1] then a is equal to `underlinebb(3/5)`.

Explanation:

We have, f(x) = `{{:((sin^3(sqrt(3)).log(1  +  3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3))  -  1)x)",", x ≠ 0),(                         a",", x = 0):}`

For continuity in [0, 1], f(0) = `lim_(x rightarrow 0) f(x)` otherwise it is discontinuous.

∴ a = `lim_(x rightarrow 0) (sin^3(sqrt(x)).log(1  +  3x))/(x(tan^-1 sqrt(x))^2.(e^(5sqrt(x))  -  1))`

= `lim_(x rightarrow 0) [3/5 . (sin^3 sqrt(x))/(sqrt(x))^3 . (sqrt(x))^3/(tan^-1 sqrt(x))^3 xx (log(1 + 3x))/(3x) . (5sqrt(x))/(3^(5sqrt(x)) - 1)]`

= `3/5 lim_(x rightarrow 0) sin^3/(sqrt(x))^3 . (sqrt(x))^3/(tan^-1 sqrt(x)) xx (log(1 + 3x))/(3x) . (5sqrt(x))/(e^(5sqrt(x)) - 1)`

= `3/5`

∴ a = `3/5`