Advertisements
Advertisements
Question
Examine the continuity of the following:
x2 cos x
Advertisements
Solution
Let f(x) = x2 cos x
f(x) is defined at 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 cos x`
= x20 cos 0
f(x0) = x20 cos 0
From equation (1) and (2), we have
`lim_(x -> x_0) x^2 cos x = f(x_0)`
∴ The limit at x = x0 exist and is equal to the value of the function f(x) at x = x0.
Since x0 is arbitrary, the limit of the function exist and is equal to the value of the function for all points in R.
∴ f(x) satisfies all conditions for continuity.
Hence f (x) is a continuous function in R.
APPEARS IN
RELATED QUESTIONS
Prove that f(x) = 2x2 + 3x - 5 is continuous at all points in R
Examine the continuity of the following:
ex tan x
Examine the continuity of the following:
`sinx/x^2`
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) = {{:(sinx",", 0 ≤ x ≤ pi/4),(cos x",", pi/4 < x < pi/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)`
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):}`
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) = (3 - sqrt(x))/(9 - x), x_0` = 9
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):}`
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:
Let the function f be defined by `f(x) = {{:(3x, 0 ≤ x ≤ 1),(-3 + 5, 1 < x ≤ 2):}`, then
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 value of `lim_(x -> "k") x - [x]`, where k is an integer 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
