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Select the correct answer from the given alternatives: If f(x) = [x] for x ∈ (–1, 2) then f is discontinuous at

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प्रश्न

Select the correct answer from the given alternatives:

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

पर्याय

  • x = –1, 0, 1, 2,

  • x = –1, 0, 1

  • x = 0, 1

  • x = 2

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उत्तर

If f(x) = [x] for x ∈ (–1, 2) then f is discontinuous at x = 0, 1

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पाठ 8: Continuity - MISCELLANEOUS EXERCISE-8 [पृष्ठ १७७]

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बालभारती Mathematics and Statistics (Arts and Science) Part 2 [English] Standard 11 Maharashtra State Board
पाठ 8 Continuity
MISCELLANEOUS EXERCISE-8 | Q (I) (10) | पृष्ठ १७७

संबंधित प्रश्‍न

Examine whether the function is continuous at the points indicated against them:
f(x) = x3 − 2x + 1,         for x ≤ 2
      = 3x − 2,                 for 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


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


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


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?


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):}`


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:

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


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.


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) `{:(= (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 x3 − 5x2 + 3x + 6 = 0 has at least two real root between x = 1 and x = 5


Let f : [-1, 2] → [0, ∞] be a continuous function such that f(x) = f(1 - x) ∀ x ∈ [-1, 2].

Let R1 = `int_-1^2 xf(x) dx` and R2 be the area of the region bounded by y = f(x), x = -1, x = 2 and the X-axis. Then, ______


If f(x) = `{(8-6x;   0<x≤2), (4x-12;    2<x≤3),(2x+10;    3<x≤6):}` then f(x) is ______ 


If f(x) = `{{:(log(sec^2 x)^(cot^2x)",", "for"  x ≠ 0),(K",", "for"  x = 0):}`

is continuous at x = 0, then K is ______.


If f(x) = `{{:(x, "for"  x ≤ 0),(0,
"for"  x > 0):}`, then f(x) at x = 0 is ______.


If f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`

is continuous at x = 0, then the value of k is ______.


If f(x) = `{{:((sin^3(sqrt(3)).log(1  +  3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3))  -  1)x)",", x ≠ 0),(                         a",", x = 0):}`

is continuous in [0, 1] then a is equal to ______.


Which condition defines continuity of a real-valued function \[f\] at \[x=c\]?


Which one-sided-limit condition is required for continuity at \[x=c\]?


If \[f\] is defined on the closed interval \[[a,b]\], which condition establishes continuity at the left endpoint \[a\]?


Which condition makes a function discontinuous at \[x=a\]?


When is \[f(x)\] non-removable discontinuous at \[x=a\]?


At which points is the reciprocal function \[f(x)=\frac{1}{x}\] continuous?


For \[f(x)=x^2\], what verifies continuity at \[x=0\]?


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