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

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प्रश्न

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

योग
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उत्तर

Every polynomial is continuous for all values of x f(x) = x3 + 2x2 − x − 2 is a polynomial and hence continuous at x = − 2.

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अध्याय 8: Continuity - EXERCISE 8.1 [पृष्ठ १७२]

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बालभारती Mathematics and Statistics (Arts and Science) Part 2 [English] Standard 11 Maharashtra State Board
अध्याय 8 Continuity
EXERCISE 8.1 | Q 1) (i) | पृष्ठ १७२

संबंधित प्रश्न

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^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) `{:(= 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 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) `{:(= 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) = `(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


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


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):}`


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


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)` =


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) = [x + 1] for x ∈ [−2, 2)

Where [*] is greatest integer function.


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 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):}`


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) = `{{:((sin5x)/(x^2 + 2x)",", x ≠ 0),(k + 1/2",", x = 0):}` is continuous at x = 0, then the value of k is ______.


Let f be the function defined by

f(x) = `{{:((x^2 - 1)/(x^2 - 2|x - 1| - 1)",", x ≠ 1),(1/2",", x = 1):}`


If the function f(x) = `[tan(π/4 + x)]^(1/x)` for x ≠ 0 is = K for x = 0 continuous at x = 0, then K = ?


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


If the function f(x) defined by

f(x) = `{{:(x sin  1/x",", "for"  x = 0),(k",", "for"  x = 0):}`

is continuous at x = 0, then k is equal to ______.


Which of the following is not continuous for all x?


The function f(x) = x – |x – x2| is ______.


For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.


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

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


`lim_(x rightarrow 0) (e^(x^2) - cosx)/x^2` is equal to ______.


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