Advertisements
Advertisements
प्रश्न
Examine the continuity of the following:
|x + 2| + |x – 1|
Advertisements
उत्तर
Let f(x) = |x + 2| + |x – 1|
f(x) is defined for all points of R.
Let x0 be an arbitrary point in R.
Then `lim_(x -> x_0)f(x) = lim_(x -> x_0) (|x + 2| + |x - 1|)`
= `|x_0 + 2| + |x_0 - 1|` .......(1)
`f(x_0) = |x_0+ 2| + |x_0 - 1|` .......(2)
From equation (1) and (2) we get
`lim_(x -> x_0)f(x) = f(x_0)`
Thus the limit of the function f(x) exist at x = x0 and is equal to the value of the function at x = x0.
Since x = x0 is an arbitrary point in R, the above
result is true for all points in R.
Hence f(x) is continuous at all points of R.
APPEARS IN
संबंधित प्रश्न
Examine the continuity of the following:
x2 cos x
Examine the continuity of the following:
ex tan x
Examine the continuity of the following:
x . log x
Examine the continuity of the following:
`|x - 2|/|x + 1|`
Find the points of discontinuity of the function f, where `f(x) = {{:(x + 2",", "if", x ≥ 2),(x^2",", "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):}`
Show that the function `{{:((x^3 - 1)/(x - 1)",", "if" x ≠ 1),(3",", "if" x = 1):}` is continuous om `(- oo, oo)`
Let `f(x) = {{:(0",", "if" x < 0),(x^2",", "if" 0 ≤ x ≤ 2),(4",", "if" x ≥ 2):}`. Graph the function. Show that f(x) continuous on `(- oo, oo)`
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) = {{:((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
Find the constant b that makes g continuous on `(- oo, oo)`.
`g(x) = {{:(x^2 - "b"^2,"if" x < 4),("b"x + 20, "if" x ≥ 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?
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:
The function `f(x) = {{:((x^2 - 1)/(x^3 + 1), x ≠ - 1),("P", x = -1):}` is not defined for x = −1. The value of f(−1) so that the function extended by this value is continuous 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
