Advertisements
Advertisements
Question
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?
Solution
`f(x) = (x^2 - 1)/(x^3 - 1)`
f(x) is not defined at x = 1
`lim_(x -> 1) f(x) = lim_(x -> 1) (x^2 - 1)/(x^3 - 1)`
= `lim_(x -> 1) ((x + 1)(x - 1))/((x - 1)(x^2 + x + 1))`
= `lim_(x -> 1) (x + 1)/(x^2 + x + 1)`
= `(1 + 1)/(1^2 + 1 + 1)`
= `2/3`
`lim_(x -> 1) f(x) = 2/3`
The function f(x) has a removable discontinuity at x = 1.
Redefine f(x) as
`f(x) = {{:((x^2 - 1)/(x^3 - 1)",", "if" x ≠ 1),(2/3",", "if" x = 1):}`
∴ f(1) = `2/3`.
Then f(x) will be continuous at x = 1
APPEARS IN
RELATED QUESTIONS
Examine the continuity of the following:
x + sin x
Examine the continuity of the following:
x . log x
Examine the continuity of the following:
`|x - 2|/|x + 1|`
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):}`
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):}`
If f and g are continuous functions with f(3) = 5 and `lim_(x -> 3) [2f(x) - g(x)]` = 4, find g(3)
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):}`
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
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.
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:
At x = `3/2` the function f(x) = `|2x - 3|/(2x - 3)` is
Choose the correct alternative:
Let f : R → R be defined by `f(x) = {{:(x, x "is irrational"),(1 - x, x "is rational"):}` then f 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
