Advertisements
Advertisements
Question
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]):}`
Advertisements
Solution
For x ∈ [0, 3), f(x) = `(x^2 + x + 1)/(x + 1)`, being a rational function is continuous except when its denominator x + 1 = 0 i.e., at x = – 1, which does not belong to [0, 3)
∴ f is continuous on [0, 3).
For x ∈ [3, 6], f(x) = `(3x + 4)/(x^2 - 5)`, being a rational function is continuous except when its denominator x2 – 5 = 0 i.e., at x = `± sqrt(5)` But `± sqrt(5) ∉ [3, 6]`
∴ f is continuous on [0, 6] except possibly at x = 3
Continuity at x = 3
f(x) = `(x^2 + x + 1)/(x + 1)`, for x ∈ [0, 3)
∴ `lim_(x -> 3^-) "f"(x) = lim_(x -> 3) (x^2 + x + 1)/(x + 1)`
= `(lim_(x -> 3)(x^2 + x + 1))/(lim_(x -> 3) (x + 1))`
= `(9 + 3 + 1)/(3 + 1)`
= `13/4`
Also, f(x) = `(3x + 4)/(x^2 - 5)`, for x ∈ [3, 6]
∴ `lim_(x -> 3^+) "f"(x) = "f"(3) = (9 + 4)/(9 - 5) = 13/4`
∴ `"f"(3) = lim_(x -> 3^+) "f"(x) = lim_(x -> 3^-) "f"(x)`
∴ f is continuous at x = 3.
Hence, f is continuous on its domain [0, 6].
APPEARS IN
RELATED QUESTIONS
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`
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) `{:(= (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.
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) `{:(=("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 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) `{:(= (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(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):}`
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:
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 ______.
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) `{:( = (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
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 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.
If f(x) `= sqrt(4 + "x" - 2)/"x", "x" ne 0` be continuous at x = 0, then f(0) = ______.
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) = `{{:(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 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 ______.
The function f(x) = x – |x – x2| is ______.
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.
If \[\mathrm{f}(x)= \begin{cases} \mathrm{m}x+1, & x\leqslant\frac{\pi}{2} \\ \\ \mathrm{sin}x+\mathrm{n}, & x>\frac{\pi}{2} & \end{cases}\], is continuous at \[x=\frac{\pi}{2},( \begin{array} {c}\mathrm{m,n\in\mathbb{Z}} \end{array})\] then
Let `f(x) = (2 - sqrt(x + 4))/(sin 2x), x ≠ 0`. In order that f(x) is continuous at x = 0, f(0) is to be defined as ______.
