Set, Relation and FunctionHard
Question
For every integer n, let an and bn be real numbers. Let function f : IR → IR be given by f(x) = 
, for all integers n. If f is continuous, then which of the following hold(s) for all n?
, for all integers n. If f is continuous, then which of the following hold(s) for all n?Options
A.an-1-bn-1 = 0
B.an-bn = 1
C.an -bn+1 = 1
D.an-1 -bn = - 1
Solution
At x = 2n
L.H.L. =
(bn + cos π (2n - h)) = bn + 1
R.H.L. = (an + sin π (2n + h)) = an
f(2n) = an
For continuity bn + 1 = an
At x = 2n + 1
L.H.L = (an + sin π (2n + 1 - h)) = an
R.H.L = (bn+1 + cos (π (2n + 1 - h))) = bn+1 - 1
f(2n + 1) = an
For continuity
an = bn+1 - 1
an-1 - bn = - 1.
L.H.L. =
(bn + cos π (2n - h)) = bn + 1R.H.L. =
(an + sin π (2n + h)) = anf(2n) = an
For continuity bn + 1 = an
At x = 2n + 1
L.H.L =
(an + sin π (2n + 1 - h)) = anR.H.L =
(bn+1 + cos (π (2n + 1 - h))) = bn+1 - 1f(2n + 1) = an
For continuity
an = bn+1 - 1
an-1 - bn = - 1.
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