Class 12 RD Sharma Solutions - Chapter 9 Continuity - Exercise 9.1 | Set 1

Question 1. Test the continuity of the following function at the origin:

Solution:

Given that

Now, let us consider LHL at x = 0

Now, let us consider RHL at x = 0

So, LHL ≠ RHL

Therefore, f(x) is discontinuous at origin and the discontinuity is of 1st kind.

Question 2. A function f(x) is defined as  . Show that f(x) is continuous at x = 3.

Solution:

Given that

So, here we check the given f(x) is continuous at x = 3,

Now, let us consider LHL at x = 3

Now, let us consider RHL at x = 3

So, f(3) = 5

LHL= RHL = f(3) 

Therefore, f(x) is continuous at x = 3

Question 3. A function f(x) is defined as

Show that f(x) is continuous at x = 3.

Solution:

Given that

So, here we check the given f(x) is continuous at x = 3,

Now, let us consider LHL at x = 3

Now, let us consider RHL at x = 3

So, f(3) = 6

LHL= RHL= f(3)

Therefore, f(x) is continuous at x = 3

Question 4. 

Find whether f(x) is continuous at x = 1

Solution:

Given that

So, here we check the given f(x) is continuous at x = 1,

Now, let us consider LHL at x = 1

Now, let us consider RHL at x = 1

So, f(1) = 2

LHL= RHL = f(1)

Therefore, f(x) is continuous at x = 1

Question 5. If 

 Find whether f(x) is continuous at x = 0.

Solution:

Given that

So, here we check the given f(x) is continuous at x = 0,

Now, let us consider LHL at x = 0

Now, let us consider RHL at x = 0

So, f(0) = 1

LHL = RHL≠ f(0)

Therefore, f(x) is discontinuous at x = 0.

Question 6. If 

Find whether f is continuous at x = 0.

Solution:

Given that

So, here we check the given f(x) is continuous at x = 0,

Now, let us consider LHL at x = 0

Now, let us consider RHL at x = 0

So, LHL≠ RHL

Therefore, the f(x) is discontinuous at x = 0. 

Question 7. Let 

Show that f(x) is discontinuous at x = 0.

Solution:

Given that

So, here we check the given f(x) is discontinuous at x = 0,

Now, let us consider LHL at x = 0

= 2 × 1/4 = 1/2                           

Now, let us consider RHL at x = 0

= 2 × 1/4 = 1/2                           

f(0) = 1

LHL= RHL ≠ f(0)

Therefore, the f(x) is discontinuous at x = 0. 

Question 8. Show that  is discontinuous at x = 0.

Solution:

Given that

So, here we check the given f(x) is discontinuous at x = 0,

Now, let us consider LHL at x = 0

Now, let us consider RHL at x = 0

f(0) = 2

Thus, LHL= RHL≠ f(0)

Therefore, f(x) is discontinuous at x = 0. 

Question 9. Show that  is discontinuous at x = a.

Solution:

Given that

So, here we check the given f(x) is discontinuous at x = a,

Now, let us consider LHL at x = a

Now, let us consider RHL at x = a

Thus, LHS ≠ RHL

Therefore, the f(x) is discontinuous at x = a.

Discuss the continuity of the following functions at the indicated points(s):

Question 10 (i). 

Solution:

Given that

So, here we check the continuity of the given f(x) at x = 0,

Let us consider LHL,

Now, let us consider RHL,

f(0) = 0

Thus, LHL= RHL= f(0) = 0

Therefore, f(x) is continuous at x = 0.

Question 10 (ii).  at x = 0

Solution:

Given that

So, here we check the continuity of the given f(x) at x = 0,

Let us consider LHL,

Now, let us consider RHL,

f(0) = 0

Thus, LHL= RHL = f(0) = 0

Therefore, f(x) is continuous at x = 0.

Question 10 (iii).  at x = a

Solution:

Given that

So, here we check the continuity of the given f(x) at x = a,

Let us consider LHL,

Now, let us consider RHL,

f(a) = 0

Thus, LHL= RHL= f(a) = 0

Therefore, f(x) is continuous at x = 0.

Question 10 (iv).  at x = 0

Solution:

Given that

So, here we check the continuity of the given f(x) at x = 0,

= 1/2 × 1/1 = 1/2                        

And, 

f(0) = 7

 ≠ f(0)

Therefore, f(x) is discontinuous at x = 0.

Question 10 (v).  n ∈ N at x = 1 

Solution:

Given that

So, here we check the continuity of the given f(x) at x = 1,

Let us consider LHL,

Now, let us consider RHL,

f(1) = n - 1

Thus, LHL = RHL ≠ f(1)

Therefore, f(x) is discontinuous at x = 1.

Question 10 (vi).  at x = 1

Solution:

Given that

So, here we check the continuity of the given f(x) at x = 1,

Let us consider LHL,

Now, let us consider RHL,

f(1) = 2

LHL= RHL = f(1) = 2

Therefore, f(x) is discontinuous at x = 1.

Question 10 (vii).  at x = 0

Solution:

Given that

So, here we check the continuity of the given f(x) at x = 0,

Let us consider LHL,

Let us consider RHL,

Thus, LHL ≠ RHL

Therefore, f(x) is discontinuous at x = 0.

Question 10 (viii).  at x = a

Solution:

Given that, 

f(x) = (x - a)sin{1/(x - a)}, x > 0

= (x - a)sin{1/(x - a)}, x < 0

= 0, x = a 

Let us consider LHL,

Now, let us consider RHL,

⇒ 

Therefore, f(x) is continuous at x = a.

Question 11. Show that is discontinuous at x = 1.

Solution:

Given that, 

So, here we check the given f(x) is discontinuous at x = 1,

Let us consider LHL,

Now, let us consider RHL,

LHL ≠ RHL

Therefore, f(x) is discontinuous at x = 1.

Question 12. Show that   is continuous at x = 0

Solution:

Given that,

So, here we check the given f(x) is continuous at x = 0,

Let us consider LHL,

Let us consider RHL,

f(0) = 3/2

Thus, LHL = RHL = f(0) = 3/2

Therefore, f(x) is continuous at x = 0.

Question 13. Find the value of 'a' for which the function f defined by 

  is continuous at x = 0.

Solution:

Given that,

Let us consider LHL,

Now, let us consider RHL,

⇒ 

= (1/2) × 1 × 1

⇒ 

If f(x) is continuous at x = 0, then

⇒ a = 1/2

Question 14. Examine the continuity of the function 

 at x = 0

Also sketch the graph of this function.

Solution:

Given that, 

So, here we check the continuity of the given f(x) at x = 0,

Let us consider LHL,

Now, let us consider RHL,

LhL ≠ RHL

So, the f(x) is discontinuous.

👁 Image

Question 15. Discuss the continuity of the function

 at the point x = 0.

Solution:

Given that, 

So, here we check the continuity of the given f(x) at x = 0,

Let us consider LHL,

Now, let us consider RHL,

f(0) = 1

LHL = RHL ≠ f(0)

Hence, the f(x) is discontinuous at x = 0. 

Summary

This chapter typically covers:

• Definition of continuity
• Types of discontinuities
• Continuity of composite functions
• Continuity in intervals
• Properties of continuous functions
• Intermediate Value Theorem