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

• Last Updated : 26 May, 2021

### 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 â‰  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

### 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

### 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

### 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 = RHLâ‰  f(0)

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

### 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, LHLâ‰  RHL

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

### 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 Ã— 1/4 = 1/2

Now, let us consider RHL at x = 0

= 2 Ã— 1/4 = 1/2

f(0) = 1

LHL= RHL â‰  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= RHLâ‰  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 â‰  RHL

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

### 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 Ã— 1/1 = 1/2

And,

f(0) = 7

â‰  f(0)

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

### Question 10 (v).  n âˆˆ 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 â‰  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 â‰  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,

â‡’

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 â‰  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.

### is continuous at x = 0.

Solution:

Given that,

Let us consider LHL,

Now, let us consider RHL,

â‡’

= (1/2) Ã— 1 Ã— 1

â‡’

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

â‡’ a = 1/2

### 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 â‰  RHL

So, the f(x) is discontinuous.

### 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 â‰  f(0)

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

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