Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
`lim_(x -> 0^+) "f"(x) = 7/3` ...(given)
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (x/(tan3x) + 2)`
= `lim_(x -> 0^-) x/(tan 3x) + lim_(x -> 0^-) 2`
= `lim_(x -> 0^-) 1/((tan3x)/x) + lim_(x -> 0^-) 2`
= `lim_(x -> 0^-) (1/((tan3x)/(3x) xx 3)) + lim_(x -> 0^-) 2`
= `(lim_(x -> 0^-) 1)/(3 lim_(x -> 0^-) ((tan 3x)/(3x))) + lim_(x -> 0^-) 2`
= `1/(3(1)) + 2`
= `7/3`
∴ `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^-) "f"(x)`
∴ f(x) is continuous at x = 0.
APPEARS IN
संबंधित प्रश्न
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^2 + 18x - 19)/(x - 1)",", "for" x ≠ 1),(= 20",", "for" x = 1):}}` at x = 1
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) `{:( =(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) = `(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`
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) = `(cos^2 x - sin^2 x - 1)/(sqrt(3x^2 + 1) - 1)` for x ≠ 0, is continuous at x = 0 then find f(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) `{:(= (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
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:
f(x) `{:(= (32^x - 8^x - 4^x + 1)/(4^x - 2^(x + 1) + 1)",", "for" x ≠ 0),(= "k""," , "for" x = 0):}` is continuous at x = 0, then value of ‘k’ is
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]):}`
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 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) = `1/(1 - x)`, the number of points of discontinuity of f{f[f(x)]} 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 ______.
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 ______.
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.
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 ______.
Which condition defines continuity of a real-valued function \[f\] at \[x=c\]?
A real function \[f\] is a continuous function when it is continuous at which points?
If \[f\] is defined on the closed interval \[[a,b]\], which condition establishes continuity at the left endpoint \[a\]?
When is \[f(x)\] non-removable discontinuous at \[x=a\]?
For which values is the constant function \[f(x)=k\] continuous?
For \[f(x)=|x|=\begin{cases}-x,&x<0\\x,&x\geq0\end{cases}\], what are the LHL, RHL, and \[f(0)\] at \[x=0\]?
For \[f(x)=\begin{cases}x^3+3,&x\ne0\\1,&x=0\end{cases}\], which statement is correct at \[x=0\]?
