Advertisements
Advertisements
प्रश्न
Examine the continuity of the following:
cot x + tan x
Advertisements
उत्तर
Let f(x) = cot x + tan x
f(x) is not defined a x= `("n"x)/2`, n ∈ z
∴ f(x) is defined for all pints of `"R" - {("n"pi)/2, "n" ∈ "z"}`
Let x0 be an arbitrary point in `"R" - {("n"pi)/2}`, n ∈ z
Then `lim_(x -> x_0) f(x) = lim_(x -> x_0) (cotx + tan x)`
= `cox_ + tan x_0` .......(1)
`f(x_0) = cot x_0 + tan x_0` .......(2)
From equation (1) and (2) we have
`lim_(x -> x_0) (cotx + tan x) = f(x_0)`
∴ The limit of the function f(x) exists at x = x0 and is equal to the value of the function f(x) at x = x0.
Since x0 is an arbitrary point , the above result is true for all points of `"R" - {("n"x)/2}`, n ∈ z.
∴ f(x) is continuous at all points of `"R" - {("n"x)/2}`, n ∈ z.
APPEARS IN
संबंधित प्रश्न
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|
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) = {{:(x^3 - 3",", "if" x ≤ 2),(x^2 + 1",", "if" x < 2):}`
Show that the function `{{:((x^3 - 1)/(x - 1)",", "if" x ≠ 1),(3",", "if" x = 1):}` is continuous om `(- oo, oo)`
For what value of `alpha` is this function `f(x) = {{:((x^4 - 1)/(x - 1)",", "if" x ≠ 1),(alpha",", "if" x = 1):}` continuous at 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):}`
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
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
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:
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 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
