Discuss the continuity of the following functions.
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x . cos x

Solution:

A function is said to be continuous when the graph of the function is a single unbroken curve.

It is known that if g and h are two continuous functions,

then g + h,g − h, and g,h is also continuous.

Let g(x) = sin x and h(x) = cos x are continuous functions.

It is evident that g(x) = sin x is defined for every real number.

Let c be a real number.

Put x = c + h 

If x→ c, then h→0

g(c) = sin c

lim_x→c g(x) = lim_x→c sin x

= lim _h→0 sin(c+h) = lim _h→0 [sin c cos h+cos c sin h] 

= lim _h→0 (sin c cos h) + lim _h→0 (cos c sin h)

= sin c cos 0 + cos c sin0

=sin c(1) + cos c(0)

= sin c

⇒ lim_x→c g(x) = g(c)

Therefore,

g(x) = sin x is a continuous function.

Let h(x) = cos x

It is evident that h(x) = cos x defined for every real number.

Let c be a real number. Put x = c + h

If x→c, then h→0

h(c) = cos c

lim_x→c h(x) = lim_{x→ c} cos x

= lim _h→0 cos(c + h) = lim _h→0 [cos c cos h−sin c sin h]

= lim _h→0 (cos c cosh) − lim _h→0 (sin c sin h)

= cos c cos0 − sin c sin0 = cos c(1) − sinc(0) = cos c

⇒ lim_x→c h(x) = h(c)

Therefore, h(x) = cos x is a continuous function.

Therefore, it can be concluded that,

(i) f(x) = g(x) + h(x) = sin x + cos x is a continuous function.

(ii) f(x) = g(x) − h(x) = sin x − cos x is a continuous function.

(iii) f(x) = g(x) × h(x) = sin x × cos x is a continuous function

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 21

Discuss the continuity of the following functions. (i) f(x) = sinx + cosx (ii) f(x) = sinx − cosx (iii) f(x) = sinx . cosx

Summary:

It can be concluded that, (i) f(x) = sin x + cos x is a continuous function. (ii) f(x) = sin x − cos x is a continuous function. (iii) f(x) = sin x × cos x is a continuous function