Answer :

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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