Advertisements
Advertisements
प्रश्न
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):}`
Advertisements
उत्तर
`(x^2 - 3x - 10)/(x - 5)` is not defined at x = 5
∴ f(x) = `(x^2 - 3x - 10)/(x - 5)` where x ∈ [3, 5) ∪ (5, 6]
We can write f(x) explicitly, as follows:
f(x) `{:(=(x^2 - 3x - 10)/(x - 5),"," 3 ≤ x < 5),(= 10, "," x = 5),(= (x^2 - 3x - 10)/(x - 5), "," 5 < x ≤ 6),(= (x^2 - 3x - 10)/(x - 5), "," 6 < x ≤ 9):}`
∵ x2 – 3x – 10 = (x – 5) (x + 2)
∴ f(x) `{:(= x + 2",", 3 < x < 5),(= 10",", x = 5),(= x + 2",", 5 < x):}`
f(5) = 10
`lim_(x -> 5^-) "f"(x) = lim_(x -> 5^-) (x + 2)` = 5 + 2 = 7
`lim_(x -> 5^+) "f"(x) = lim_(x -> 5^+) (x + 2)` = 5 + 2 = 7
∴ f(5) = `lim_(x -> 5) "f"(x)`
∴ f(x) is continuous on its domain except at x = 5
APPEARS IN
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 the continuity of f(x) = x3 + 2x2 − x − 2 at x = − 2
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
Discuss the continuity of the function f(x) = |2x + 3|, at x = `(−3)/(2)`
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`
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) = `(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?
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 x3 − 3x = 0 between 1 and 2.
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) = [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) `{:(= 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 + 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]):}`
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
If f(x) is continuous at x = 3, where
f(x) = ax + 1, for x ≤ 3
= bx + 3, for x > 3 then.
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 function `f(x)={((x^2-9)/(x-3), ",when "xne3),(k, ",when "x =3):}` is continuous at x = 3, then the value of k will be ______.
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 ______.
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 ______.
`lim_(x rightarrow 0) (e^(x^2) - cosx)/x^2` is equal to ______.
