Advertisements
Advertisements
प्रश्न
Examine the continuity of the following:
ex tan x
Advertisements
उत्तर
Let f(x) = ex tan x
f(x) is defined at ail points of R.
Expect at `(2"n" + 1) pi/2`, n ∈ Z.
Let x0 be an arbitrary point in `"R" - (2"n" + 1) pi/2`, n ∈ Z
Then `lim_(x -> x_0) f(x) = lim_(x -> x_0) "e"^x tan x`
= `"e"^(x_0) tan x_0` .......(1)
`f(x_0) = "e"^(x_0) tan x_0` .......(2)
From equation (1) and (2) we get
`lim_(x -> x_0) "e"^x tan x = f(x_0)`
∴ 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. f(x) exists at all points in `"R" - (2"n" + 1) pi/2`, n ∈ Z and is equal to the value of the function f(x) at that points.
∴ f(x) satisfies all conditions for continuity.
Hence, f(x) is continuous at all points of `"R" - (2"n" + 1) pi/2`, n ∈ Z
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:
x . log x
Examine the continuity of the following:
|x + 2| + |x – 1|
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):}`
Show that the function `{{:((x^3 - 1)/(x - 1)",", "if" x ≠ 1),(3",", "if" x = 1):}` is continuous om `(- 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):}`
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):}`
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.
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:
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:
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
