Advertisements
Advertisements
प्रश्न
Examine the continuity of the following:
x + sin x
उत्तर
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
संबंधित प्रश्न
Prove that f(x) = 2x2 + 3x - 5 is continuous at all points in R
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:
`sinx/x^2`
Examine the continuity of the following:
`(x^2 - 16)/(x + 4)`
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 = 1, `f(x) = {{:((x^2 - 1)/(x - 1)",", x ≠ 1),(2",", x = 1):}`
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)`
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):}`
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?
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?
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:
At x = `3/2` the function f(x) = `|2x - 3|/(2x - 3)` is
