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NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.1 mathematics part 1

EXERCISE 5.1


Page No 159:

Question 1:

Prove that the functionis continuous at

Answer:

Therefore, f is continuous at x = 0

Therefore, is continuous at x = −3

Therefore, f is continuous at x = 5

Question 2:

Examine the continuity of the function.

Answer:

Thus, f is continuous at x = 3

Question 3:

Examine the following functions for continuity.

(a)

 (b)

(c)  (d) 

Answer:

(a) The given function is

It is evident that f is defined at every real number k and its value at k is k − 5.

It is also observed that, 

Hence, f is continuous at every real number and therefore, it is a continuous function.

(b) The given function is

For any real number k ≠ 5, we obtain

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.

(c) The given function is

For any real number c ≠ −5, we obtain

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.

(d) The given function is 

This function f is defined at all points of the real line.

Let c be a point on a real line. Then, c < 5 or c = 5 or c > 5

Case I: c < 5

Then, (c) = 5 − c

Therefore, f is continuous at all real numbers less than 5.

Case II : c = 5

Then, 

Therefore, is continuous at x = 5

Case III: c > 5

Therefore, f is continuous at all real numbers greater than 5.

Hence, f is continuous at every real number and therefore, it is a continuous function.

Question 4:

Prove that the function is continuous at x = n, where n is a positive integer.

Answer:

The given function is f (x) = xn

It is evident that f is defined at all positive integers, n, and its value at n is nn.

Therefore, is continuous at n, where n is a positive integer.

Question 5:

Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?

Answer:

The given function f is 

At x = 0,

It is evident that f is defined at 0 and its value at 0 is 0.

Therefore, f is continuous at x = 0

At x = 1,

is defined at 1 and its value at 1 is 1.

The left hand limit of f at x = 1 is,

The right hand limit of at x = 1 is,

Therefore, f is not continuous at x = 1

At = 2,

is defined at 2 and its value at 2 is 5.

Therefore, f is continuous at = 2

Question 6:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

It is evident that the given function f is defined at all the points of the real line.

Let c be a point on the real line. Then, three cases arise.

(i) c < 2

(ii) c > 2

(iii) c = 2

Case (i) c < 2

Therefore, f is continuous at all points x, such that x < 2

Case (ii) c > 2

Therefore, f is continuous at all points x, such that x > 2

Case (iii) c = 2

Then, the left hand limit of at x = 2 is,

The right hand limit of f at x = 2 is,

It is observed that the left and right hand limit of f at x = 2 do not coincide.

Therefore, f is not continuous at x = 2

Hence, x = 2 is the only point of discontinuity of f.

Question 7:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < −3

Case II:

Therefore, f is continuous at x = −3

Case III:

Therefore, f is continuous in (−3, 3).

Case IV:

If c = 3, then the left hand limit of at x = 3 is,

The right hand limit of at x = 3 is,

It is observed that the left and right hand limit of f at x = 3 do not coincide.

Therefore, f is not continuous at x = 3

Case V:

Therefore, f is continuous at all points x, such that x > 3

Hence, x = 3 is the only point of discontinuity of f.

Question 8:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

It is known that,

Therefore, the given function can be rewritten as

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x < 0

Case II:

If c = 0, then the left hand limit of at x = 0 is,

The right hand limit of at x = 0 is,

It is observed that the left and right hand limit of f at x = 0 do not coincide.

Therefore, f is not continuous at x = 0

Case III:

Therefore, f is continuous at all points x, such that x > 0

Hence, x = 0 is the only point of discontinuity of f.

Question 9:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

It is known that,

Therefore, the given function can be rewritten as

Let c be any real number. Then, 

Also,

Therefore, the given function is a continuous function.

Hence, the given function has no point of discontinuity.

Question 10:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < 1

Case II:

The left hand limit of at x = 1 is,

The right hand limit of at x = 1 is,

Therefore, f is continuous at x = 1

Case III:

Therefore, f is continuous at all points x, such that x > 1

Hence, the given function has no point of discontinuity.

Question 11:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < 2

Case II:

Therefore, f is continuous at x = 2

Case III:

Therefore, f is continuous at all points x, such that x > 2

Thus, the given function f is continuous at every point on the real line.

Hence, has no point of discontinuity.

Question 12:

Find all points of discontinuity of f, where f is defined by

Answer:

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < 1

Case II:

If c = 1, then the left hand limit of f at x = 1 is,

The right hand limit of f at = 1 is,

It is observed that the left and right hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.

Question 13:

Is the function defined by

a continuous function?

Answer:

The given function is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < 1

Case II:

The left hand limit of at x = 1 is,

The right hand limit of f at = 1 is,

It is observed that the left and right hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.

Page No 160:

Question 14:

Discuss the continuity of the function f, where f is defined by

Answer:

The given function is

The given function is defined at all points of the interval [0, 10].

Let c be a point in the interval [0, 10].

Case I:

Therefore, f is continuous in the interval [0, 1).

Case II:

The left hand limit of at x = 1 is,

The right hand limit of f at = 1 is,

It is observed that the left and right hand limits of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

Therefore, f is continuous at all points of the interval (1, 3).

Case IV:

The left hand limit of at x = 3 is,

The right hand limit of f at = 3 is,

It is observed that the left and right hand limits of f at x = 3 do not coincide.

Therefore, f is not continuous at x = 3

Case V:

Therefore, f is continuous at all points of the interval (3, 10].

Hence, is not continuous at = 1 and = 3

Question 15:

Discuss the continuity of the function f, where f is defined by

Answer:

The given function is

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < 0

Case II:

The left hand limit of at x = 0 is,

The right hand limit of f at = 0 is,

Therefore, f is continuous at x = 0

Case III:

Therefore, f is continuous at all points of the interval (0, 1).

Case IV:

The left hand limit of at x = 1 is,

The right hand limit of f at = 1 is,

It is observed that the left and right hand limits of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case V:

Therefore, f is continuous at all points x, such that x > 1

Hence, is not continuous only at = 1

Question 16:

Discuss the continuity of the function f, where f is defined by

Answer:

The given function f is

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < −1

Case II:

The left hand limit of at x = −1 is,

The right hand limit of f at = −1 is,

Therefore, f is continuous at x = −1

Case III:

Therefore, f is continuous at all points of the interval (−1, 1).

Case IV:

The left hand limit of at x = 1 is,

The right hand limit of f at = 1 is,

Therefore, f is continuous at x = 2

Case V:

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.

Question 17:

Find the relationship between a and b so that the function f defined by

is continuous at = 3.

Answer:

The given function f is

If f is continuous at x = 3, then

Therefore, from (1), we obtain

Therefore, the required relationship is given by,

Question 18:

For what value of is the function defined by

continuous at x = 0? What about continuity at x = 1?

Answer:

The given function f is

If f is continuous at x = 0, then

Therefore, there is no value of λ for which f is continuous at x = 0

At x = 1,

f (1) = 4x + 1 = 4 × 1 + 1 = 5

Therefore, for any values of λ, f is continuous at x = 1

Question 19:

Show that the function defined by is discontinuous at all integral point. Here denotes the greatest integer less than or equal to x.

Answer:

The given function is

It is evident that g is defined at all integral points.

Let n be an integer.

Then,

The left hand limit of at x = n is,

The right hand limit of f at n is,

It is observed that the left and right hand limits of f at x = n do not coincide.

Therefore, f is not continuous at x = n

Hence, g is discontinuous at all integral points.

Question 20:

Is the function defined by continuous at =

π?

Answer:

The given function is

It is evident that f is defined at =

π.

Therefore, the given function f is continuous at = π

Question 21:

Discuss the continuity of the following functions.

(a) f (x) = sin x + cos x

(b) f (x) = sin x − cos x

(c) f (x) = sin x × cos x

Answer:

It is known that if and are two continuous functions, then

are also continuous.

It has to proved first that g (x) = sin and h (x) = cos x are continuous functions.

Let (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

Therefore, g is a continuous function.

Let h (x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c

Therefore, h is a continuous function.

Therefore, it can be concluded that

(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function

(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function

(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function

Question 22:

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

Answer:

It is known that if and are two continuous functions, then

It has to be proved first that g (x) = sin and h (x) = cos x are continuous functions.

Let (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let be a real number. Put x = c + h

If x

 c, then h

0

Therefore, g is a continuous function.

Let h (x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x ® c, then h ® 0

(c) = cos c

Therefore, h (x) = cos x is continuous function.

It can be concluded that,

Therefore, cosecant is continuous except at np, Î Z

Therefore, secant is continuous except at 

Therefore, cotangent is continuous except at np, Î Z

Question 23:

Find the points of discontinuity of f, where

Answer:

The given function f is

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

Therefore, f is continuous at all points x, such that x < 0

Case II:

Therefore, f is continuous at all points x, such that x > 0

Case III:

The left hand limit of f at x = 0 is,

The right hand limit of f at x = 0 is,

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at all points of the real line.

Thus, f has no point of discontinuity.

Question 24:

Determine if f defined by

is a continuous function?

Answer:

The given function f is

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

Therefore, f is continuous at all points ≠ 0

Case II:

 

⇒-x2≤x2sin1x≤x2 

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function.

Page No 161:

Question 25:

Examine the continuity of f, where f is defined by

Answer:

The given function f is

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

Therefore, f is continuous at all points x, such that x ≠ 0

Case II:

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function.

Question 26:

Find the values of so that the function f is continuous at the indicated point.

Answer:

The given function f is

The given function f is continuous at, if f is defined at and if the value of the f at  equals the limit of f at.

It is evident that is defined at and

Therefore, the required value of k is 6.

Question 27:

Find the values of so that the function f is continuous at the indicated point.

Answer:

The given function is

The given function f is continuous at x = 2, if f is defined at x = 2 and if the value of f at x = 2 equals the limit of f at x = 2

It is evident that is defined at x = 2 and

Therefore, the required value of.

Question 28:

Find the values of so that the function f is continuous at the indicated point.

Answer:

The given function is

The given function f is continuous at x = p, if f is defined at x = p and if the value of f at x = p equals the limit of f at x = p

It is evident that is defined at x = p and

Therefore, the required value of

Question 29:

Find the values of so that the function f is continuous at the indicated point.

Answer:

The given function is

The given function f is continuous at x = 5, if f is defined at x = 5 and if the value of f at x = 5 equals the limit of f at x = 5

It is evident that is defined at x = 5 and

Therefore, the required value of

 

Question 30:

Find the values of a and b such that the function defined by

is a continuous function.

Answer:

The given function is

It is evident that the given function f is defined at all points of the real line.

If f is a continuous function, then f is continuous at all real numbers.

In particular, f is continuous at = 2 and = 10

Since f is continuous at = 2, we obtain

Since f is continuous at = 10, we obtain

On subtracting equation (1) from equation (2), we obtain

8a = 16

⇒ a = 2

By putting a = 2 in equation (1), we obtain

2 × 2 + b = 5

⇒ 4 + b = 5

⇒ b = 1

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.

 

Question 31:

Show that the function defined by f (x) = cos (x2) is a continuous function.

Answer:

The given function is (x) = cos (x2)

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, where g (x) = cos x and h (x) = x2

It has to be first proved that (x) = cos x and h (x) = x2 are continuous functions.

It is evident that g is defined for every real number.

Let c be a real number.

Then, g (c) = cos c

Therefore, g (x) = cos x is continuous function.

h (x) = x2

Clearly, h is defined for every real number.

Let k be a real number, then h (k) = k2

Therefore, h is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, is a continuous function.

Question 32:

Show that the function defined by is a continuous function.

Answer:

The given function is

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, where

It has to be first proved that  are continuous functions.

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

Therefore, g is continuous at all points x, such that x < 0

Case II:

Therefore, g is continuous at all points x, such that x > 0

Case III:

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

(x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c

Therefore, h (x) = cos x is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, is a continuous function.

Question 33:

Examine that  is a continuous function.

Answer:

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, where

It has to be proved first that  are continuous functions.

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

Therefore, g is continuous at all points x, such that x < 0

Case II:

Therefore, g is continuous at all points x, such that x > 0

Case III:

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

(x) = sin x

It is evident that h (x) = sin x is defined for every real number.

Let be a real number. Put x = c + k

If x → c, then k → 0

(c) = sin c

Therefore, h is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, is a continuous function.

Question 34:

Find all the points of discontinuity of defined by.

Answer:

The given function is

The two functions, g and h, are defined as

Then, f = − h

The continuity of g and is examined first.

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

Therefore, g is continuous at all points x, such that x < 0

Case II:

Therefore, g is continuous at all points x, such that x > 0

Case III:

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

Clearly, h is defined for every real number.

Let be a real number.

Case I:

Therefore, h is continuous at all points x, such that x < −1

Case II:

Therefore, h is continuous at all points x, such that x > −1

Case III:

Therefore, h is continuous at x = −1

From the above three observations, it can be concluded that h is continuous at all points of the real line.

g and h are continuous functions. Therefore, g − is also a continuous function.

Therefore, has no point of discontinuity.


 

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