# Find the value of k so that the function f is continuous at the indicated point:

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 (h0)

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

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