Advertisements
Advertisements
Question
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):}`
Advertisements
Solution
`"f"(pi/4) = 3/sqrt(2)`
= `lim_(x -> pi/4) "f"(x) = lim_(x -> pi/4) ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)`
= (sin x + cos x)3 = `[(sin x + cos x)^2]^(3/2)`
= `(1 + sin 2x)^(3/2)`
∴ `lim_(x -> pi/4) "f"(x) = lim_(x -> pi/4) ((1 + sin 2x)^(3/2) - 2^(3/2))/(sin 2x - 1)`
Put 1 + sin 2x = t
∴ sin 2x = t – 1
As `x -> pi/4"," "t" -> 1 + sin 2(pi/4)`
i.e. `"t" -> 1 + sin pi/(2)`
i.e. t → 1 + 1
i.e. t → 2
∴ `lim_(x -> pi/4) "f"(x) = lim_("t" -> 2) ("t"^(3/2) - 2^(3/2))/("t" - 1 - 1)`
= `lim_("t" -> 2) ("t"^(3/2) - 2^(3/2))/("t" - 2)`
= `3/2(2)^(1/2) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `(3sqrt(2))/(2)`
= `3/sqrt(2)`
∴ `lim_(x -> pi/4) "f"(x) = "f"(pi/4)`
∴ f(x) is continuous at x = `pi/(4)`
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
Find all the point of discontinuities of f(x) = [x] on the interval (− 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) `{:( =(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
Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :
f(x) = `(1 - cos2x)/sinx`, for x ≠ 0
Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :
f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for 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
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
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
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):}`
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) = `(cos4x - cos9x)/(1 - cosx)`, for x ≠ 0
f(0) = `68/15`, at x = 0 on `- pi/2 ≤ x ≤ pi/2`
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(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):}`
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 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
Solve using intermediate value theorem:
Show that 5x − 6x = 0 has a root in [1, 2]
If f(x) = `{((x^4 - 1/81)/(x^3 - 1/27), x ≠ 1/3), (k, x = 1/3):}` is continuous at x = `1/3`, then the value of k is ______
If f(x) = `{:{(tan^-1|x|; "when" x ≠ 0), (pi/4; "when" x = 0):}`, then ______
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 f(x) = `{{:((sin5x)/(x^2 + 2x)",", x ≠ 0),(k + 1/2",", x = 0):}` is continuous at x = 0, then the value of k 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 ?
The function f(x) = x – |x – x2| 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 ______.
Which is a necessary condition for continuity at \[x=c\]?
When is \[f(x)\] removable discontinuous at \[x=a\]?
When is \[f(x)\] non-removable discontinuous at \[x=a\]?
What is the graphical meaning of a function being continuous at a point?
For \[f(x)=\begin{cases}1,&x\leq0\\2,&x>0\end{cases}\], why is \[f\] discontinuous at \[x=0\]?
Which statement about polynomial functions is correct?
For a piecewise function, at which points should continuity be checked especially?
At which points is the greatest integer function \[f(x)=[x]\] discontinuous?
