Advertisements
Advertisements
Question
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
Advertisements
Solution
f(2) = – 24 …(given)
`lim_(x -> 2) "f"(x) = lim_(x -> 2) (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))`
= `lim_(x -> 2) (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2)) xx (sqrt(x + 2) + sqrt(3x + 2))/(sqrt(x + 2) + sqrt(3x - 2))`
= `lim_(x -> 2) ((x^3 - 8)(sqrt(x + 2) + sqrt(3x - 2)))/((x + 2) - (3x - 2))`
= `lim_(x -> 2) ((x^3 - 2^3)(sqrt(x + 2) + sqrt(3x - 2)))/(-2x + 4)`
= `lim_(x -> 2) ((x - 2)(x^2 + 2x + 4)(sqrt(x + 2) + sqrt(3x - 2)))/(-2(x - 2))`
= `lim_(x -> 2) ((x^2 + 2x + 4)(sqrt(x + 2) + sqrt(3x - 2)))/(-2) ...[(becausex -> 2"," x ≠ 2),(therefore x - 2 ≠ 0)]`
= `(-1)/(2) lim_(x -> 2) (x^2 + 2x + 4) (sqrt(x + 2) + sqrt(3x - 2))`
= `(-1)/(2) lim_(x -> 2) (x^2 + 2x + 4) lim_(x -> 2) (sqrt(x + 2) + sqrt(3x - 2))`
= `(-1)/(2) xx [2^2 + 2(2) + 4] xx (sqrt(2 + 2) + sqrt(3(2) - 2))`
= `(-1)/(2) xx 12 xx (2 + 2)`
= – 24
∴ `lim_(x -> 2) "f"(x)` = f(2)
∴ f(x) is continuous at x = 2
APPEARS IN
RELATED QUESTIONS
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) {:(= sin x",", "for" x ≤ pi/4), (= cos x",", "for" x > pi/4):}} "at" x = pi/4`
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) `{:(=(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) `{:(= log_((1 + 3x)) (1 + 5x)",", "for" x > 0),(=(32^x - 1)/(8^x - 1)",", "for" x < 0):}}` at x = 0
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
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?
For what values of a and b is the function
f(x) `{:(= (x^2 - 4)/(x - 2)",", "for" x < 2),(= "a"x^2 - "b"x + 3",", "for" 2 ≤ x < 3),(= 2x - "a" + "b"",", "for" x ≥ 3):}}` continuous for every x on R?
Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for" -3 ≤ x ≤ 2),(= |x - 5|",", "for" 2 < x ≤ 7):}`.
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:
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) = `(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 ______.
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) `{:(= (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):}`
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 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
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) = `{(8-6x; 0<x≤2), (4x-12; 2<x≤3),(2x+10; 3<x≤6):}` then f(x) is ______
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) = `1/(1 - x)`, the number of points of discontinuity of f{f[f(x)]} is ______.
If f(x) = `{{:(x, "for" x ≤ 0),(0,
"for" x > 0):}`, then f(x) at x = 0 is ______.
If f(x) = `{{:((x - 4)/(|x - 4|) + a",", "for" x < 4),(a + b",", "for" x = 4 "is continuous at" x = 4","),((x - 4)/(|x - 4|) + b",", "for" x > 4):}`
then ______.
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.
Which condition defines continuity of a real-valued function \[f\] at \[x=c\]?
If \[f\] is defined on the closed interval \[[a,b]\], which condition establishes continuity at the left endpoint \[a\]?
Which sequence correctly checks continuity at \[x=c\]?
For which values is the constant function \[f(x)=k\] continuous?
For a piecewise function, at which points should continuity be checked especially?
What does the greatest integer function \[f(x)=[x]\] give?
For \[f(x)=x^2\], what verifies continuity at \[x=0\]?
For \[f(x)=|x|=\begin{cases}-x,&x<0\\x,&x\geq0\end{cases}\], what are the LHL, RHL, and \[f(0)\] at \[x=0\]?
