Given,

…(1)

We need to find the value of k such that f(x) is continuous at x = 0.

A function f(x) is said to be continuous at x = c if,

Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).

Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)

Now, let’s assume that f(x) is continuous at x = 0.

∴

As we have to find k so pick out a combination so that we get k in our equation.

In this question we take LHL = f(0)

∴

⇒ {using equation 1}

∵ cos(-x) = cos x and sin(-x) = - sin x

∴

Also, 1 – cos x = 2 sin² (x/2)

∴

As this limit can be evaluated directly by putting value of h because it is taking indeterminate form(0/0)

So we use sandwich or squeeze theorem according to which –

⇒ 2

Dividing and multiplying by (kh/2)² to match the form in formula we have-

⇒

Using algebra of limits we get –

⇒

Applying the formula-

⇒ 1 × (k²/4) = (1/4)

⇒ k² = 1

⇒ (k+1)(k – 1) = 0

∴ k = 1 or k = -1