RBSE Solutions for Class 12 Maths Chapter 6 Continuity and Differentiability Ex 6.1

Rajasthan Board RBSE Class 12 Maths Chapter 6 Continuity and Differentiability Ex 6.1

RBSE Solutions For Class 12 Maths Chapter 6 Question 1.
Examine the following functions for continuity :
RBSE Solutions For Class 12 Maths Chapter 6 Continuity And Differentiability
Solution:
(a) Given function.
RBSE Solutions For Class 12 Maths Chapter 6.1 Continuity And Differentiability
Left hand limit

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Right hand limit

RBSE Class 12 Maths Chapter 6 Continuity And Differentiability
RBSE Solutions For Class 12 Maths Chapter 6.1 Continuity And Differentiability
Left hand limit
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Right hand limit

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Hence, given function is not continuous at x = 0.
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and at x = 3
f (3) = 1 + x ⇒ 1 + 3 = 4
∴ f (3 – 0) = f(3 + 0) = f(3) = 4
Hence, given function is continuous at x = 3.

(d) Given function,
RBSE Solution Class 12 Maths Continuity And Differentiability
Left hand limit
f(0 – 0) = limh→0 f(0 – h)
= limh→0 sin (0 – h)
= limh→0 sin (- h)
= limh→0 (-sin h)
[∵ sin (-θ) = -sin θ]
= 0
Right hand limit
RBSE Solution Class 12th Maths Continuity And Differentiability
Hence, given function is continuous at x = 0.

(e) Given function,

RBSE Solutions For Class 12 Maths Continuity And Differentiability
Left hand limit
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Right hand limit
12th Class RBSE Solution Continuity And Differentiability
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Hence, given function is not continuous at x = 0.

(f) Given function,
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Left hand limit
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Right hand limit
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Hence, given function is not continuous at x = a.

(g) Given function,
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Left hand limit
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Right hand limit

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Hence, given function is continuous at x = a.

RBSE Solutions For Class 12 Maths Chapter 6.1  Question 2.
Test the continuity of f(x) at x = 3 if f(x) = x – [x].
Solution:
Given function f(x) = x – [x], at x = 3
Left hand limit
f (3 – 0) = limh→0 f(3 – h)
= limh→0 (3 – h) – [3 – h]
= 3 – 2
= 1
[∵ 2 is just before greatest integer 3]

Right hand limit
f (3 + 0) = limh→0 f(3 + h)
= limh→0 (3 + h) – [3 + h]
= 3 – 3
= 0
∴ f (3 – 0) ≠ f (3 + 0)
Hence, given function at not continuous at x = 3.

Question 3.
RBSE Solutions For Class 12th Maths Continuity And Differentiability
continuous at x = 2, then find k.
Solution :
Left hand limit
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Right hand limit
RBSE Solution For Class 12th Continuity And Differentiability

RBSE Solutions For Class 12 Maths Chapter 6 Miscellaneous Question 4.
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the continuity of f (x) is closed interval [- 1, 2].
Solution:
Here, we will test the continuity of function at x = 0 and 0 ∈ [-1,2].
Left hand limit
f(0 – 0) = limh→0 f(0 – h)
= limh→0 (0 – h)2
= limh→0 h2
= 0

Right hand limit
f (0 + 0) = limh→0 f(0 + h)
= limh→0 4(0 + h) – 3
= limh→0 Ah – 3
= 0 – 3
= – 3
∴ f (0 – 0) ≠ f(0 + 0)
LHL ≠ RHL
So, function is not continuous at x = 0 and x ∈ [- 1, 2]
At x = 1
Left hand limit
f(1 – h) = limh→0 4(1 – h) – 3
= limh→0 4 – 3 – Ah
=4 – 3 – 0 = 1

Right hand limit
f(1 + h) = limh→0 5(1 + h)2 – 4 (1 + h)
= limh→0 5(1 + h2 + 2h) – (4 + Ah)
= limh→0 5h2 +10h + 5 – 4 – 4h
= 5 × 0 + 10 × 0 + 1 – 4(0)
= 1
Value of function at x = 1
f(1) = 4 × 1 – 3 = 1
∵ limh→0 f(1 – A) = f(1) = limh→0 f(1 + h)
∴Function is continuous at x = 1.
Hence, function is not continuous in interval [-1, 2],

RBSE Solutions for Class 12 Maths