Advertisements
Advertisements
Question
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
Advertisements
Solution
The function f(x) is not defined at x = – 4.
`f(x) = (x^3 + 4^3)/(x + 4)`
`f(x) = ((x + 4)(x^2 - 4x + 16))/(x + 4)`
`f(x) = x^2 - 4x + 16`
`lim_(x -> - 4) f(x) = lim_(x -> - 4) (x^2 - 4x + 16)`
= `(- 4)^2 - 4 xx - 4 + 16`
= 16 + 16 + 16
`lim_(x -> - 4) f(x)` = 48
Limit the function f(x) exist at x = – 4.
∴ The function f(x) has a removable discontinuity at x = – 4.
Redefine the function f (x) as
`g(x) = {{:((x^3 + 64)/(x + 4), "if" x ≠ - 4),(48, "if" x = - 4):}`
Clearly, the function g(x) is continuous on R.
APPEARS IN
RELATED QUESTIONS
Examine the continuity of the following:
x2 cos x
Examine the continuity of the following:
ex tan x
Examine the continuity of the following:
e2x + x2
Examine the continuity of the following:
`sinx/x^2`
Examine the continuity of the following:
`(x^2 - 16)/(x + 4)`
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) = {{:(4x + 5",", "if", x ≤ 3),(4x - 5",", "if", x > 3):}`
Find the points of discontinuity of the function f, where `f(x) = {{:(x + 2",", "if", x ≥ 2),(x^2",", "if", x < 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):}`
Consider the function `f(x) = x sin pi/x`. What value must we give f(0) in order to make the function continuous everywhere?
The function `f(x) = (x^2 - 1)/(x^3 - 1)` is not defined at x = 1. What value must we give f(1) inorder to make f(x) continuous at x =1?
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:
At x = `3/2` the function f(x) = `|2x - 3|/(2x - 3)` is
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
