Advertisements
Advertisements
प्रश्न
At the given point x0 discover whether the given function is continuous or discontinuous citing the reasons for your answer:
x0 = 3, `f(x) = {{:((x^2 - 9)/(x - 3)",", "if" x ≠ 3),(5",", "if" x = 3):}`
Advertisements
उत्तर
`lim_(x -> 3^-) f(x) = lim_(x -> 3^-) (x^2 - 9)/(x - 3)`
= `lim_(x -> 3^-) ((x + 3)(x - 3))/(x - 3)`
= `lim_(x -> 3^-) (x + 3)`
= 3 + 3
= 6 .........(1)
`lim_(x -> 3^+) f(x) = lim_(x -> 3^+) (x^2 - 9)/(x - 3)`
= `lim_(x -> 3^+) ((x + 3)(x - 3))/(x - 3)`
= `lim_(x -> 3^+) (x + 3)`
= 3 + 3
= 6 .........(2)
From equations (1) and (2)
`lim_(x -> 3^-) f(x) = lim_(x -> 3^-) f(x)` = 6
∴ `lim_(x -> 3) f(x)` = 6 ........(3)
`f(3)` = 5 ........(4)
From equations (3) and (4) we have
`lim_(x -> 3) f(x) ≠ f(3)`
∴ f(x) is not continuous at x0 = 3.
APPEARS IN
संबंधित प्रश्न
Examine the continuity of the following:
e2x + x2
Examine the continuity of the following:
x . log x
Examine the continuity of the following:
`sinx/x^2`
Examine the continuity of the following:
cot x + tan x
Find the points of discontinuity of the function f, where `f(x) = {{:(x^3 - 3",", "if" x ≤ 2),(x^2 + 1",", "if" x < 2):}`
At the given point x0 discover whether the given function is continuous or discontinuous citing the reasons for your answer:
x0 = 1, `f(x) = {{:((x^2 - 1)/(x - 1)",", x ≠ 1),(2",", x = 1):}`
Show that the function `{{:((x^3 - 1)/(x - 1)",", "if" x ≠ 1),(3",", "if" x = 1):}` is continuous om `(- oo, oo)`
Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.
`f(x) = {{:(2x + 1",", "if" x ≤ - 1),(3x",", "if" - 1 < x < 1),(2x - 1",", "if" x ≥ 1):}`
Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.
`f(x) = {{:((x - 1)^3",", "if" x < 0),((x + 1)^3",", "if" x ≥ 0):}`
A function f is defined as follows:
`f(x) = {{:(0, "for" x < 0;),(x, "for" 0 ≤ x ≤ 1;),(- x^2 +4x - 2, "for" 1 ≤ x ≤ 3;),(4 - x, "for" x ≥ 3):}`
Is the function continuous?
Which of the following functions f has a removable discontinuity at x = x0? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R.
`f(x) = (x^2 - 2x - 8)/(x + 2), x_0` = – 2
Which of the following functions f has a removable discontinuity at x = x0? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R.
`f(x) = (x^3 + 64)/(x + 4), x_0` = – 4
Which of the following functions f has a removable discontinuity at x = x0? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R.
`f(x) = (3 - sqrt(x))/(9 - x), x_0` = 9
Consider the function `f(x) = x sin pi/x`. What value must we give f(0) in order to make the function continuous everywhere?
State how continuity is destroyed at x = x0 for the following graphs.
Choose the correct alternative:
Let the function f be defined by `f(x) = {{:(3x, 0 ≤ x ≤ 1),(-3 + 5, 1 < x ≤ 2):}`, then
Choose the correct alternative:
At x = `3/2` the function f(x) = `|2x - 3|/(2x - 3)` is
Choose the correct alternative:
Let a function f be defined by `f(x) = (x - |x|)/x` for x ≠ 0 and f(0) = 2. Then f is
