Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 8 - Continuity [Latest edition]

Examine the continuity of f(x) = x³ + 2x² − x − 2 at x = − 2

Examine the continuity of `"f"(x)  {:(= sin x",",  "for"  x ≤ pi/4), (= cos x",",  "for"  x > pi/4):}}  "at"  x = pi/4`

Examine the continuity of `f(x) = {:((x^2 - 9)/(x  - 3)",",  "for"  x ≠ 3),(=8",",  "for"  x = 3):}}` at x = 3.

Examine whether the function is continuous at the points indicated against them:

f(x)  `{:(= x^3 - 2x + 1",",  "if"  x ≤ 2),(= 3x - 2",",  "if"  x > 2):}}` at x = 2

Examine whether the function is continuous at the points indicated against them :

f(x) `{:( = (x^2 + 18x - 19)/(x - 1)",",  "for"  x ≠ 1),(= 20",",  "for"  x = 1):}}` at x = 1

Examine whether the function is continuous at the points indicated against them :

f(x) `{:(= x/(tan3x) + 2",",   "for"  x < 0),(= 7/3",",  "for"  x ≥ 0):}}  "at"  x = 0`

Find all the point of discontinuities of f(x) = [x] on the interval (− 3, 2).

Discuss the continuity of the function f(x) = |2x + 3|, at x = `(−3)/(2)`

Test the continuity of the following function at the point or interval indicated against them :

f(x)  `{:(= (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)",",  "for"  x ≠ 2),(= 1/5",",  "for"  x = 2):}}`at x = 2

Test the continuity of the following function at the point or interval indicated against them :

f(x)  `{:(= (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))",",  "for"  x ≠ 2),(= -24",",  "for"  x = 2):}}` at x = 2

Test the continuity of the following function at the point or interval indicated against them :

f(x) `{:(= 4x + 1",",  "for"  x ≤  8/3),(= (59 - 9x)/3 ",",  "for"  x > 8/3):}}  "at"  x = 8/3`

Test the continuity of the following function at the point or interval indicated against them :

f(x) `{:(= ((27 - 2x)^(1/3) - 3)/(9 - 3(243 + 5x)^(1/5))",",  "for"  x ≠ 0),(= 2",",  "for"  x = 0):}}` at x = 0.

Test the continuity of the following function at the point or interval indicated against them:

f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",",  "for"  0 < x < 3","  x ≠ 2),(= 12",",  "for"  x = 2),(= (2 - 2x - x^2)/(x - 4)",",  "for"  3 ≤ x < 4):}}` at x = 2

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) = `(x^2 - 10x + 21)/(x - 7)`

Identify the discontinuity for the following function as either a jump or a removable discontinuity.

f(x) `{:(= x^2 + 3x - 2",",  "for"  x ≤ 4),(= 5x + 3",",  "for"  x > 4):}`

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) `{:(= x^2 - 3x - 2",",  "for"  x < -3),(= 3 + 8x",",  "for"  x > -3):}`

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) `{:(= 4 + sin x",",  "for"  x < pi),(= 3 - cos x",",  "for"  x > pi):}`

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(1 - cos2x)/sinx`, for x ≠ 0

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for x ≠ 0

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(x^2 - 1)/(x^3 + 1)` for x ≠ – 1

Discuss the continuity of the following function at the point indicated against them :

f(x) = `{:(=( sqrt(3) - tanx)/(pi - 3x)",", x ≠ pi/3),(= 3/4",", x = pi/3):}}  "at"  x = pi/3`

Discuss the continuity of the following function at the point indicated against them :

f(x)  `{:(=("e"^(1/x) - 1)/("e"^(1/x) + 1)",",  "for"  x ≠ 0),(= 1",", "for"  x = 0):}}` at x = 0

Discuss the continuity of the following function at the point indicated against them :

f(x)  `{:(=(4^x - 2^(x + 1) + 1)/(1 - cos 2x)",",  "for"  x ≠ 0),(= (log 2)^2/2",",  "for"  x = 0):}}` at x = 0.

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it becomes continuous :

f(x) `{:(=("e"^(5sinx) - "e"^(2x))/(5tanx - 3x)",",   "for"  x ≠ 0),(= 3/4",",   "for"  x = 0):}}` at x = 0

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :

f(x) `{:(= log_((1 + 3x)) (1 + 5x)",", "for"  x > 0),(=(32^x - 1)/(8^x - 1)",",  "for"  x < 0):}}` at x = 0

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :

f(x) = `((3 - 8x)/(3 - 2x))^(1/x)`, for x ≠ 0

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :

f(x) `{:(= 3x + 2",",  "for"  -4 ≤ x ≤-2),(= 2x - 3";",  "for"  -2 < x ≤ 6):}`

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :

f(x) `{:(= (x^3 - 8)/(x^2 - 4)",",  "for"  x > 2),(= 3",",  "for"  x = 2),(= ("e"^(3(x - 2)^2 - 1))/(2(x - 2)^2) ",",  "for"  x < 2):}`

If f(x) = `(sqrt(2 + sin x) - sqrt(3))/(cos^2x), "for"  x ≠ pi/2`, is continuous at x = `pi/2` then find `"f"(pi/2)`

If f(x) = `(cos^2 x - sin^2 x - 1)/(sqrt(3x^2 + 1) - 1)` for x ≠ 0, is continuous at x = 0 then find f(0)

If f(x) = `(4^(x - π) + 4^(π - x) - 2)/(x - π)^2` for x ≠ π, is continuous at x = π, then find f(π).

If f(x) `{:(= (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)",",  "for"  x ≠ 0), (= "k"",",  "for"  x = 0):}}` is continuous at x = 0, find k

If f(x)  `{:(= (5^x + 5^(-x) - 2)/(x^2)"," , "for"  x ≠ 0),(= k",",  "for"  x = 0):}}` is continuous at x = 0, find k

If f(x) `{:(= (sin2x)/(5x) - "a"",", "for"  x > 0),(= 4 ",", "for"  x = 0),(= x^2 + "b" - 3",", "for"  x < 0):}}` is continuous at x = 0, find a and b

For what values of a and b is the function

f(x) `{:(= "a"x + 2"b" + 18",",  "for"  x ≤ 0),(= x^2 + 3"a" - "b"",",  "for"  0 < x ≤ 2),(= 8x - 2",",  "for"  x > 2):}}` continuous for every x?

For what values of a and b is the function

f(x) `{:(= (x^2 - 4)/(x - 2)",", "for"  x < 2),(= "a"x^2 - "b"x + 3",", "for"  2 ≤ x < 3),(= 2x - "a" + "b"",", "for"  x ≥ 3):}}` continuous for every x on R?

Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for"  -3 ≤ x ≤ 2),(= |x - 5|",", "for"  2 < x ≤ 7):}`.

Discuss the continuity of f(x) at x = `pi/4` where, 

f(x) `{:(= ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)",", "for"  x ≠ pi/4),(= 3/sqrt(2)",", "for"  x = pi/4):}`

Determine the values of p and q such that the following function is continuous on the entire real number line.

f(x) `{:(= x + 1",", "for"   1 < x < 3),(= x^2 + "p"x + "q"",", "for"  |x - 2| ≥ 1):}`

Show that there is a root for the equation 2x³ − x − 16 = 0 between 2 and 3.

Show that there is a root for the equation x³ − 3x = 0 between 1 and 2.

Let f(x) = ax + b (where a and b are unknown)

= x² + 5 for x ∈ R

Find the values of a and b, so that f(x) is continuous at x = 1

Suppose f(x) `{:(= "p"x + 3",", "for"  "a" ≤ x ≤ "b"),(= 5x^2 − "q"",", "for"  "b" < x ≤ "c"):}`

Find the condition on p, q, so that f(x) is continuous on [a, c], by filling in the blanks.

f(b) = ______

`lim_(x -> "b"^+) "f"(x)` = _______

∴ pb + 3 = _______ − q

∴ p = `"_____"/"b"` is the required condition

Select the correct answer from the given alternatives:

f(x) = `{:(= (2^(cotx) - 1)/(pi - 2x)",", "for"  x ≠ pi/2),(= log sqrt(2)",", "for"  x = pi/2):}`

Select the correct answer from the given alternatives:

If f(x) = `(1 - sqrt(2) sinx)/(pi - 4x), "for"  x ≠ pi/4` is continuous at x = `pi/4`, then `"f"(pi/4)` =

Select the correct answer from the given alternatives:

If f(x) = `((sin2x)tan5x)/("e"^(2x) - 1)^2`, for x ≠ 0 is continuous at x = 0, then f(0) is

Select the correct answer from the given alternatives:

f(x) = `(x^2 - 7x + 10)/(x^2 + 2x - 8)`, for x ∈ [– 6, – 3]

Select the correct answer from the given alternatives:

If f(x) `{:(= "a"x^2 + "b"x + 1",", "for"  |x −1| ≥ 3), (= 4x + 5",", "for"  -2 < x < 4):}` is continuous everywhere then,

Select the correct answer from the given alternatives:

f(x) `{:(= ((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))",", "for"  x ≠ 0),(= "k"",", "for"  x = 0):}` is continuous at x = 0, then ‘k’ =

Select the correct answer from the given alternatives:

f(x) `{:(= (32^x - 8^x - 4^x + 1)/(4^x - 2^(x + 1) + 1)",", "for"  x ≠ 0),(= "k""," , "for"  x = 0):}` is continuous at x = 0, then value of ‘k’ is

Select the correct answer from the given alternatives:

If f(x) = `(12^x - 4^x - 3^x + 1)/(1 - cos 2x)`, for x ≠ 0 is continuous at x = 0 then the value of f(0) is ______.

Select the correct answer from the given alternatives:

If f(x) = `((4 + 5x)/(4 - 7x))^(4/x)`, for x ≠ 0 and f(0) = k, is continuous at x = 0, then k is

Select the correct answer from the given alternatives:

If f(x) = [x] for x ∈ (–1, 2) then f is discontinuous at

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= (x^2 - 3x - 10)/(x - 5)",", "for"  3 ≤ x ≤ 6","  x ≠ 5),(= 10",", "for"  x = 5),(=(x^2 - 3x - 10)/(x - 5)",", "for"  6 < x ≤ 9):}`

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= 2x^2 - 2x + 5",", "for"  0 ≤ x ≤ 2),(= (1 - 3x - x^2)/(1 - x) "," , "for"  2 < x < 4),(= (x^2 - 25)/(x - 5)",", "for"  4 ≤ x ≤ 7 and x ≠ 5),(= 7",", "for"  x = 5):}`

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) = `(cos4x - cos9x)/(1 - cosx)`, for x ≠ 0

f(0) = `68/15`, at x = 0 on `- pi/2 ≤ x ≤ pi/2`

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:( = (sin^2pix)/(3(1 - x)^2) ",", "for"  x ≠ 1),(= (pi^2sin^2((pix)/2))/(3 + 4cos^2 ((pix)/2)) ",", "for"  x = 1):}}` at x = 1

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= (|x + 1|)/(2x^2 + x - 1)",", "for"  x ≠ -1),(= 0",", "for"  x = -1):}}` at x = – 1

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) = [x + 1] for x ∈ [−2, 2)

Where [*] is greatest integer function.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= 2x^2 + x + 1",", "for"  |x - 3| ≥ 2),(= x^2 + 3",", "for"  1 < x < 5):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= x^2 + x - 3,","  "for"  x ∈ [ -5, -2)),(= x^2 - 5,","  "for"  x ∈ (-2, 5]):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= x^2 + 5x + 1"," , "for"  0 ≤ x ≤ 3),(= x^3 + x + 5"," , "for"  3 < x ≤ 6):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= (x^2 + x + 1)/(x + 1)"," , "for"  x ∈ [0, 3)),(=(3x +4)/(x^2 - 5)"," , "for"  x ∈ [3, 6]):}`

Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:

f(x) = `((x + 3)(x^2 - 6x + 8))/(x^2 - x - 12)`

Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:

f(x) `{:(= x^2 + 2x + 5"," , "for"  x ≤ 3),( = x^3 - 2x^2 - 5",", "for"  x > 3):}`

Find k if following function is continuous at the point indicated against them:

f(x) `{:(= ((5x - 8)/(8 - 3x))^(3/(2x - 4))",", "for"  x ≠ 2),(= "k"",", "for"  x = 2):}}` at x = 2

Find k if following function is continuous at the point indicated against them:

f(x) `{:(= (45^x - 9^x - 5^x + 1)/(("k"^x - 1)(3^x - 1))",", "for"  x ≠ 0),(= 2/3",", "for"  x = 0):}}` at x = 0

Find a and b if following function is continuous at the point or on the interval indicated against them:

f(x) `{:(= (4tanx + 5sinx)/("a"^x - 1)",", "for"  x < 0),(= (9)/(log2)",", "for"  x = 0),(= (11x + 7x*cosx)/("b"^x - 1)",", "for"  x > 0):}`

Find a and b if following function is continuous at the point or on the interval indicated against them:

f(x) `{:(= "a"x^2 + "b"x + 1",", "for"  |2x - 3| ≥ 2),(= 3x + 2",", "for"  1/2 < x < 5/2):}`

Find f(a), if f is continuous at x = a where,

f(x) = `(1 + cos(pi x))/(pi(1 - x)^2)`, for x ≠ 1 and at a = 1

Find f(a), if f is continuous at x = a where,

f(x) = `(1 - cos[7(x - pi)])/(5(x - pi)^2`, for x ≠ π at a = π

Solve using intermediate value theorem:

Show that 5^x − 6x = 0 has a root in [1, 2]

Solve using intermediate value theorem:

Show that x³ − 5x² + 3x + 6 = 0 has at least two real root between x = 1 and x = 5