Well-structured Class 12 ISC Maths Solutions Chapter 5 Continuity and Differentiability Ex 5.3 facilitate a deeper understanding of mathematical principles.

## ML Aggarwal Class 12 Maths Solutions Section A Chapter 5 Continuity and Differentiability Ex 5.3

Question 1.

Is tan x a continuous function.

Solution:

Let f(x) = tan x ;

D_{f} = R – {odd multiple of \(\frac{\pi}{2}\)}

Consider c ∈ D_{f} be any real number

∴ c be any real number except odd multiple of \(\frac{\pi}{2}\)

∴ f(c) = tan c

Now, \(\underset{x \rightarrow 0}{\mathrm{Lt}}\) f(x) = \(\underset{x \rightarrow 0}{\mathrm{Lt}}\) tan x

= tan c

= f(c)

∴ f is continuous at x = c

but c be any arbitrary element of D_{f}

Thus f is continuous in its domain.

Hence f is a continuous function.

Question 2.

State which of the following functions are continuous ?

(i) sin x + cos x

(ii) sin x – cos x

(iii) sin x . cos x.

Solution:

(i) Given f(x) = sin x + cos x

since sin x and cos x is continuous everywhere.

∴ sin x + cos x is also continuous everywhere as the sum of two continuous functions is a continuous function.

(ii) again sin x and cos x continuous everywhere.

∴ sin x – cos x is also continuous everywhere as the difference of two continuous functions is continuous.

(iii) again sin x and cos x continuous everywhere.

∴ f (x) = sin x cos x is also continuous everywhere, since the product of two continuous functions is also a continuous function.

Question 3.

Is the functin of defined by f (x) = 2 – 3x + |x| a continuous function?

Solution:

Let f(x) = 2 – 3x,

which is a polynomial in x and hence continuous for all x ∈ R.

and g(x) = |x|, which is continuous for all x ∈ R.

We know that, if f is continuous on domain D_{1} and g is continuous on domain D_{2}

Then f + g is continuous on D_{1} ∩ D_{2}

∴ f(x) + g(x) = 2 – 3x + |x| is continuous on R ∩ R = R

∴ 2 – 3x + |x| is a continuous functin.

Question 4.

Is the functin f defined by f(x) = |sin x| a continuous function? (NCERT)

Solution:

Given f(x) = |sin x|

Since sin x is continuous for all x ∈ R.

∴ | sin x | is also continuous for all x ∈ R.

[∵ If f is continuous on domain D.

Then |f| is also continuous on D]

Thus | sin x | is a continuous function.

Question 5.

Is the function f defined by f(x) = cos |x| a continuous function ? (NCERT)

Solution:

Let g (x) = x is continuous for all x ∈ R

and h (x) = cos x is continuous for all x ∈ R

Thus composite function hog is also continuous ∀ X ∈ R

[∵ if g is continuous at x = c and h is continuous at g (c).

Then hog is continuous at c]

But (hog) (x) = h {g (x)} = h { |x| } = cos |x|

Thus cos | x | is continuous for all x ∈ R

Hence f (x) = cos | x | be a continuous function.

Question 6.

Prove that the following functions are continuous :

(i) \(\frac{2 x^3-7 x^2+3}{(x-1)(x+3)}\)

(ii) | sec x + tan x|

Solution:

Let f(x) = \(\frac{2 x^3-7 x^2+3}{(x-1)(x+3)}\)

D_{f} = R – {1, – 3}

Since every rational function is continuous at every point of its domain.

Here f be a rational function.

∴ f be continuous at every point of its domain.

∴ f be a continuous function.

(ii) Let f (x) = sec x, which is a continuous function

and g (x) = tan x, which is also a continuous function.

Since sum of two continuous functions is continuous function.

∴ f(x) + g(x) = sec x + tan x is also a continuous function.

∴ | sec x + tan x | is also a continuous function.

[∵ If f is continuous on domain D then |f| is also continuous on domain D].

Question 7.

Examine the following functions for continuity :

(i) cos (x^{2}) (NCERT)

(ii) sin (3x^{2} – 5)

(iii) tan^{-1} (2x^{2} + 3). (NCERT)

Solution:

(i) Given f(x) = cos x^{2}

Let a be any arbitrary real number.

L.H.L. = \(\underset{x \rightarrow a^{-}}{\mathrm{Lt}}\) f(x)

= \(\underset{h \rightarrow 0}{\mathrm{Lt}}\) f (a – h)

= \(\underset{h \rightarrow 0}{\mathrm{Lt}}\) cos (a – h)^{2}

= cos a^{2}

and R.H.L. = \(\underset{x \rightarrow a^{+}}{\mathrm{Lt}}\) f(x)

= \(\underset{h \rightarrow 0}{\mathrm{Lt}}\) f (a + h)

= \(\underset{h \rightarrow 0}{\mathrm{Lt}}\) cos (a + h)^{2}

= cos a^{2}

also f(a) = cos a^{2}

∴ LH.L = R.H.L = f(a) = cos a^{2}

Hence f(x) is continuous at x = a.

Since a be any arbitrary real number.

Thus f(x) = cos x^{2} is continuous everywhere.

(ii) Let f(x) = 3x^{2} – 5, which is a polynomial in x

and hence continuous for all x ∈ R.

and let g (x) = sin x, which is continuous for all x ∈ R

Thus the composite function gof is continuous for all x ∈ R.

[∵ If f is continuous at x = c and g is continuous at f(c). Then gof is continuous at c]

But (gof) (x) = g (f(x))

= g (3x^{2} – 5)

= sin (3x^{2} – 5)

Thus, sin (3x^{2} – 5) is continuous for all x ∈ R.

Thus, sin (3x^{2} – 5) be a continuous function.

(iii) Let f(x) = 3x^{2} – 5,

which is a polynomial function and hence continuous for all x ∈ R.

and let g (x) = tan^{-1} x,

which is continuous function ∀ X ∈ R.

Thus, the composite function gof is continuous ∀ x ∈ R

[ If f is continuous at c and g is continuous at f(c). Then g of is continuous at c]

But (gof) (x) = g [f (x)] = g(3x^{2} – 5) = tan^{-1} (3x^{2} – 5)

Thus tan^{-1} (3x^{2} – 5) is continuous for all x ∈ R

Hence tan^{-1} (3x^{2} – 5) is a continuous function.