Advertisements
Advertisements
Question
Find the points of discontinuity of the function f, where `f(x) = {{:(4x + 5",", "if", x ≤ 3),(4x - 5",", "if", x > 3):}`
Solution
`lim_(x -> 3) f(x) = lim_(x -> 3^-) (4x + 5)`
= 4 × 3 + 5
`lim_(x -> 3) f(x)` = 17 .........(1)
`lim_(x -> 3) f(x) = lim_(x -> 3^+) (4x - 5)`
= 4 × 3 – 5
`lim_(x -> 3) f(x)` = 12 – 5 = 7 .........(2)
From equations (1) and (2) we have
`lim_(x -> 3) f(x) ≠ lim_(x -> 3^+) f(x)`
∴ `lim_(x -> 3) f(x)` does not exist.
Hence f(x) is not continuous at x = 3.
∴ x = 3 is the point of dicontinuity.
APPEARS IN
RELATED QUESTIONS
Examine the continuity of the following:
ex tan x
Examine the continuity of the following:
`sinx/x^2`
Examine the continuity of the following:
|x + 2| + |x – 1|
Examine the continuity of the following:
`|x - 2|/|x + 1|`
Find the points of discontinuity of the function f, where `f(x) = {{:(x^3 - 3",", "if" x ≤ 2),(x^2 + 1",", "if" x < 2):}`
Find the points of discontinuity of the function f, where `f(x) = {{:(sinx",", 0 ≤ x ≤ pi/4),(cos x",", pi/4 < x < pi/2):}`
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):}`
Show that the function `{{:((x^3 - 1)/(x - 1)",", "if" x ≠ 1),(3",", "if" x = 1):}` is continuous om `(- oo, oo)`
For what value of `alpha` is this function `f(x) = {{:((x^4 - 1)/(x - 1)",", "if" x ≠ 1),(alpha",", "if" x = 1):}` continuous at 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) = {{:(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):}`
State how continuity is destroyed at x = x0 for the following graphs.
State how continuity is destroyed at x = x0 for the following graphs.
Choose the correct alternative:
If f : R → R is defined by `f(x) = [x - 3] + |x - 4|` for x ∈ R then `lim_(x -> 3^-) f(x)` is equal to
Choose the correct alternative:
Let f be a continuous function on [2, 5]. If f takes only rational values for all x and f(3) = 12, then f(4.5) is equal to
