Question

Let α ∈ R be such that the function

f(x) = `{{:((cos^-1(1 - {x}^2)sin^-1(1 - {x}))/({x} - {x}^3)",", x ≠ 0),(α",", x = 0):}`

is continuous at x = 0, where {x} = x – [x], [x] is the greatest integer less than or equal to x.

Options

• No such α exists

• α = `π/sqrt(2)`

• α = 0

• α = `π/4`

Answer 1

No such α exists

Explanation:

We know that when x `rightarrow` 0^–, then {x} = 1 – h where h `rightarrow` 0

LHL = `lim_(h rightarrow 0) (cos^-1(1 - (1 - h)^2)sin^-1(h))/((1 - h)(1 - (1 - h)^2)`

= `lim_(h rightarrow 0) (cos^-1(1 - (1 - h)^2)sin^-1(h))/((1 - h)h(2 - h)`

= `(π/2 xx 1)/(1 xx 2)`

= `π/4`

and when x `rightarrow` 0⁺ then {x} = h where h `rightarrow` 0

RHL = `lim_(h rightarrow 0) (cos^-1(1 - h^2)sin^-1(1 - h))/(h - h^3)`

= `lim_(h rightarrow 0) (sin^-1sqrt(1 - (1 - h^2)^2) sin^-1(1 - h))/(h(1 - h^2))`

= `lim_(h rightarrow 0) (sin^-1(hsqrt(2 - h^2)) sin^-1(1 - h))/(h(1 - h^2))`

= `lim_(h rightarrow 0) (sin^-1(hsqrt(2 - h^2)))/(hsqrt(2 - h^2)) xx (hsqrt(2 - h^2))/(h(1 - h^2)) xx sin^-1(1 - h)`

= `1 xx sqrt(2)/1 xx π/2`

= `π/sqrt(2)`

LHL ≠ RHL

So, f(x) is discontinuous at x = 0.