Advertisements
Advertisements
Question
Examine the continuity of the following:
x + sin x
Advertisements
Solution
Let f(x) = sin x
f(x) is defined at all points of R.
Let x0 be an arbitrary point in R.
`lim_(x -> x_0) f(x) = lim_(x -> x_0) (x + sin x)`
= xo + sin x0 ........(1)
f(xo) = xo + sin xo ........(2)
From equations (1) and (2) we get
`lim_(x -> x_0) f(x) = f(x_0)`
∴ At all points of R, the limit of f(x) exists and is equal to the value of the function.
Thus, f(x) satisfies ail conditions for continuity.
Therefore, f(x) is continuous at all points of f(x).
APPEARS IN
RELATED QUESTIONS
Examine the continuity of the following:
x2 cos x
Examine the continuity of the following:
e2x + x2
Examine the continuity of the following:
x . log x
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:
cot x + tan x
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^3 - 3",", "if" x ≤ 2),(x^2 + 1",", "if" x < 2):}`
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) = {{:(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^3 + 64)/(x + 4), x_0` = – 4
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:
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 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
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
