Question

The value of k (k < 0) for which the function f defined as

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

is continuous at x = 0 is:

Options

• ±1

• −1

• `±1/22`

• `1/2`

Answer 1

−1

Explanation:

`lim/("x" -> 0) ((1-cos"kx")/("x"sin"x")) = 1/2`

`lim/("x" -> 0) ((2 sin ((2"kx")/2))/("x" sin"x")) = 1/2`

`lim/("x" -> 0) 2 ("k"/2)^2 ((sin ("kx"/2))/("kx"/2))^2 ("x"/sin"x") = 1/2`

k² = 1 ⇒ k = ±1 but k < 0 ⇒ k = −1