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

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 D₁ and g is continuous on domain D₂
Then f + g is continuous on D₁ ∩ D₂
∴ 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 (x²) (NCERT)
(ii) sin (3x² – 5)
(iii) tan^-1 (2x² + 3). (NCERT)
Solution:
(i) Given f(x) = cos x²
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 a²

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 a²
also f(a) = cos a²
∴ LH.L = R.H.L = f(a) = cos a²
Hence f(x) is continuous at x = a.
Since a be any arbitrary real number.
Thus f(x) = cos x² is continuous everywhere.

(ii) Let f(x) = 3x² – 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 (3x² – 5)
= sin (3x² – 5)
Thus, sin (3x² – 5) is continuous for all x ∈ R.
Thus, sin (3x² – 5) be a continuous function.

(iii) Let f(x) = 3x² – 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(3x² – 5) = tan^-1 (3x² – 5)
Thus tan^-1 (3x² – 5) is continuous for all x ∈ R
Hence tan^-1 (3x² – 5) is a continuous function.