Q. 195.0( 2 Votes )

Show that the function f(x) = |sin x + cos x| is continuous at x = .

Answer :

Given,


f(x) = |sin x + cos x| …(1)


We need to prove that f(x) is continuous at x = π


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 according to above theory-


f(x) is continuous at x = π if -



Clearly,


LHL =


LHL {using eqn 1}


sin (π – x) =sin x & cos (π – x) = - cos x


LHL =


LHL = | sin 0 – cos 0 | = |0 – 1|


LHL = 1 …(2)


Similarly, we proceed for RHL-


RHL =


RHL {using eqn 1}


sin (π + x) = -sin x & cos (π + x) = - cos x


RHL =


RHL = | - sin 0 – cos 0 | = |0 – 1|


RHL = 1 …(3)


Also, f(π) = |sin π + cos π| = |0 – 1| = 1 …(4)


Clearly from equation 2, 3 and 4 we can say that



f(x) is continuous at x = π …proved


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