Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
`lim_(x -> 2^-) "f"(x) = lim_(x -> 2^-) (x^3 - 2x + 1)`
= (2)3 – 2(2) + 1
= 5
`lim_(x -> 2^+) "f"(x) = lim_(x -> 2^+) (3x - 2)`
= 3(2) – 2
= 4
∴ `lim_(x -> 2^-) "f"(x) ≠ lim_(x -> 2^+) "f"(x)`
∴ f(x) is discontinuous at x = 2
APPEARS IN
संबंधित प्रश्न
Examine whether the function is continuous at the points indicated against them:
f(x) = x3 − 2x + 1, for x ≤ 2
= 3x − 2, for 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 the continuity of `f(x) = {:((x^2 - 9)/(x - 3)",", "for" x ≠ 3),(=8",", "for" x = 3):}}` at x = 3.
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`
Identify the discontinuity for the following function as either a jump or a removable discontinuity.
f(x) `{:(= x^2 + 3x - 2",", "for" x ≤ 4),(= 5x + 3",", "for" x > 4):}`
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) `{:(= (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) = `(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
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(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):}`
Show that there is a root for the equation 2x3 − x − 16 = 0 between 2 and 3.
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) = `((sin2x)tan5x)/("e"^(2x) - 1)^2`, for x ≠ 0 is continuous at x = 0, then f(0) is
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
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
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= (|x + 1|)/(2x^2 + x - 1)",", "for" x ≠ -1),(= 0",", "for" x = -1):}}` at x = – 1
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 + 3)(x^2 - 6x + 8))/(x^2 - x - 12)`
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):}`
If f(x) = `[tan (pi/4 + x)]^(1/x)`, x ≠ 0 at
= k, x = 0 is continuous x = 0. Then k = ______.
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) = `{{:(tanx/x + secx",", x ≠ 0),(2",", x = 0):}`, then ______.
If f(x) = `1/(1 - x)`, the number of points of discontinuity of f{f[f(x)]} 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) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`
is continuous at x = 0, then the value of k is ______.
Which one-sided-limit condition is required for continuity at \[x=c\]?
Which condition makes a function discontinuous at \[x=a\]?
Which sequence correctly checks continuity at \[x=c\]?
What is the graphical meaning of a function being continuous at a point?
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 \[x=0\]?
