Question

If a function f defined by

`f(x) = {{:((k cos x)/(π - 2x)",", if x ≠π/2),(3, if x = π/2):}`

is continuous at `x = π/2`, then the value of k is ______.

Options

• 2

• 3

• 6

• –6

Answer 1

If a function f defined by

`f(x) = {{:((k cos x)/(π - 2x)",", if x ≠π/2),(3, if x = π/2):}`

is continuous at `x = π/2`, then the value of k is 6.

Explanation:

Since, f(x) is continuous at `x = π/2`

Therefore, `lim_(x rightarrow π/2) f(x) = f(π/2)`

`\implies lim_(x rightarrow π/2) (k cos x)/(π - 2x) = 3`

`\implies k lim_(x rightarrow π/2) (sin(π/2 - x))/(2(π/2 - x)) = 3`

`\implies k/2 lim_(x rightarrow π/2) (sin(π/2 - x))/((π/2 - x)) = 3`

`\implies k/2 xx 1 = 3`

`\implies` k = 6