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

Question

Philoid · Accepted 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 (h→0)

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