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

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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