Advertisements
Advertisements
Question
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):}`
Advertisements
Solution
f(x) `{:(= x^2 - 3x - 2",", x < -3),(= 3 + 8x",", x > -3):}`
f(x) is a polynomial function for both the intervals.
∴ f(x) is continuous for both the given intervals.
Let us test the continuity at x = − 3
`lim_(x -> -3^-) "f"(x) = lim_(x -> -3^-) (x^2 - 3x - 2)`
= (− 3)2 − 3(− 3) − 2
= 9 + 9 − 2
= 16
`lim_(x -> -3^+) "f"(x) = lim_(x -> -3^+) (3 + 8x)`
= 3 + 8(− 3)
= − 21
∴ `lim_(x -> -3^-) "f"(x) ≠ lim_(x -> -3^+) "f"(x)`
∴ `lim_(x -> -3) "f"(x)` does not exist.
∴ f(x) is discontinuous at x = − 3
∴ f(x) has a jump discontinuity at x =− 3
APPEARS IN
RELATED QUESTIONS
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 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 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
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) `{:(= 4x + 1",", "for" x ≤ 8/3),(= (59 - 9x)/3 ",", "for" x > 8/3):}} "at" x = 8/3`
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
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) = `(4^(x - π) + 4^(π - x) - 2)/(x - π)^2` for x ≠ π, is continuous at x = π, then find f(π).
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?
Show that there is a root for the equation 2x3 − x − 16 = 0 between 2 and 3.
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:
f(x) = `(x^2 - 7x + 10)/(x^2 + 2x - 8)`, for x ∈ [– 6, – 3]
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:
If f(x) = `((4 + 5x)/(4 - 7x))^(4/x)`, for x ≠ 0 and f(0) = k, is continuous at x = 0, then k is
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) `{:(= 2x^2 + x + 1",", "for" |x - 3| ≥ 2),(= x^2 + 3",", "for" 1 < x < 5):}`
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)`
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 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):}`
If f(x) = `[tan (pi/4 + x)]^(1/x)`, x ≠ 0 at
= k, x = 0 is continuous x = 0. Then k = ______.
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) = `{{:(tanx/x + secx",", x ≠ 0),(2",", x = 0):}`, then ______.
If f(x) = `1/(1 - x)`, the number of points of discontinuity of f{f[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 ______.
For what value of k, the function defined by
f(x) = `{{:((log(1 + 2x)sin^0)/x^2",", "for" x ≠ 0),(k",", "for" x = 0):}`
is continuous at x = 0 ?
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` 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 ______.
