Advertisements
Advertisements
प्रश्न
If f(x) = `(sqrt(2 + sin x) - sqrt(3))/(cos^2x), "for" x ≠ pi/2`, is continuous at x = `pi/2` then find `"f"(pi/2)`
Advertisements
उत्तर
f is given to be continuous at x = `pi/2`
∴ by defination,
`"f"(pi/2) = lim_(x -> pi/2) "f"(x)`
= `lim_(x -> pi/2) (sqrt(2 + sinx) - sqrt(3))/(cos^2x)`
= `lim_(x -> pi/2) (sqrt(2 + sinx) - sqrt(3))/(cos^2x) xx (sqrt(2 + sinx) + sqrt(3))/(sqrt(2 + sinx) + sqrt(3))`
= `lim_(x -> pi/2) ((2 + sin x) - 3)/((1 - sin^2x)(sqrt(2 + sin x) + sqrt(3))`
= `lim_(x -> pi/2) (-(1 - sin x))/((1 - sin x)(1 + sin x)(sqrt(2 + sin x) + sqrt(3))`
= `lim_(x -> pi/2) (-1)/((1 + sin x)[sqrt(2 + sinx) + sqrt(3)]) ...[because x -> pi/2, x ≠ pi/2 therefore sin x ≠ sin pi/2 = 1 therefore 1 - sin x ≠ 0]`
= `(lim_(x -> pi/2) ( - 1))/([lim_(x -> pi/2) (1 + sin x)] xx [lim_(x -> pi/2) (sqrt(2 + sin x) + sqrt(3)]`
= `(-1)/((1 + sin pi/2) (sqrt(2 + sin pi/2) + sqrt(3))`
= `(-1)/((1 + 1)(sqrt(2 + 1) + sqrt(3))`
∴ `"f"(pi/2) = (-1)/(4sqrt(3))`
APPEARS IN
संबंधित प्रश्न
Examine whether the function is continuous at the points indicated against them:
f(x) = x3 − 2x + 1, for x ≤ 2
= 3x − 2, for x > 2, at x = 2
Examine whether the function is continuous at the points indicated against them:
f(x) = `(x^2 + 18x - 19)/(x - 1)` for x ≠ 1
= 20 for x = 1, at x = 1
Examine the continuity of `"f"(x) {:(= sin x",", "for" x ≤ pi/4), (= cos x",", "for" x > pi/4):}} "at" x = pi/4`
Examine whether the function is continuous at the points indicated against them :
f(x) `{:(= x/(tan3x) + 2",", "for" x < 0),(= 7/3",", "for" x ≥ 0):}} "at" x = 0`
Discuss the continuity of the function f(x) = |2x + 3|, at x = `(−3)/(2)`
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))",", "for" x ≠ 2),(= -24",", "for" x = 2):}}` at x = 2
Test the continuity of the following function at the point or interval indicated against them:
f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",", "for" 0 < x < 3"," x ≠ 2),(= 12",", "for" x = 2),(= (2 - 2x - x^2)/(x - 4)",", "for" 3 ≤ x < 4):}}` at x = 2
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) `{:(= 4 + sin x",", "for" x < pi),(= 3 - cos x",", "for" x > pi):}`
Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :
f(x) = `(1 - cos2x)/sinx`, for x ≠ 0
Discuss the continuity of the following function at the point indicated against them :
f(x) `{:(=(4^x - 2^(x + 1) + 1)/(1 - cos 2x)",", "for" x ≠ 0),(= (log 2)^2/2",", "for" x = 0):}}` at x = 0.
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) = `((3 - 8x)/(3 - 2x))^(1/x)`, for x ≠ 0
If f(x) = `(cos^2 x - sin^2 x - 1)/(sqrt(3x^2 + 1) - 1)` for x ≠ 0, is continuous at x = 0 then find f(0)
Suppose f(x) `{:(= "p"x + 3",", "for" "a" ≤ x ≤ "b"),(= 5x^2 − "q"",", "for" "b" < x ≤ "c"):}`
Find the condition on p, q, so that f(x) is continuous on [a, c], by filling in the blanks.
f(b) = ______
`lim_(x -> "b"^+) "f"(x)` = _______
∴ pb + 3 = _______ − q
∴ p = `"_____"/"b"` is the required condition
Select the correct answer from the given alternatives:
f(x) = `{:(= (2^(cotx) - 1)/(pi - 2x)",", "for" x ≠ pi/2),(= log sqrt(2)",", "for" x = pi/2):}`
Select the correct answer from the given alternatives:
If f(x) `{:(= "a"x^2 + "b"x + 1",", "for" |x −1| ≥ 3), (= 4x + 5",", "for" -2 < x < 4):}` is continuous everywhere then,
Select the correct answer from the given alternatives:
If f(x) = `(12^x - 4^x - 3^x + 1)/(1 - cos 2x)`, for x ≠ 0 is continuous at x = 0 then the value of f(0) is ______.
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= 2x^2 - 2x + 5",", "for" 0 ≤ x ≤ 2),(= (1 - 3x - x^2)/(1 - x) "," , "for" 2 < x < 4),(= (x^2 - 25)/(x - 5)",", "for" 4 ≤ x ≤ 7 and x ≠ 5),(= 7",", "for" x = 5):}`
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) = `(cos4x - cos9x)/(1 - cosx)`, for x ≠ 0
f(0) = `68/15`, at x = 0 on `- pi/2 ≤ x ≤ pi/2`
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= x^2 + x - 3,"," "for" x ∈ [ -5, -2)),(= x^2 - 5,"," "for" x ∈ (-2, 5]):}`
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= (x^2 + x + 1)/(x + 1)"," , "for" x ∈ [0, 3)),(=(3x +4)/(x^2 - 5)"," , "for" x ∈ [3, 6]):}`
Find f(a), if f is continuous at x = a where,
f(x) = `(1 - cos[7(x - pi)])/(5(x - pi)^2`, for x ≠ π at a = π
If f(x) = `{((x^4 - 1/81)/(x^3 - 1/27), x ≠ 1/3), (k, x = 1/3):}` is continuous at x = `1/3`, then the value of k is ______
If f(x) is continuous at x = 3, where
f(x) = ax + 1, for x ≤ 3
= bx + 3, for x > 3 then.
If f(x) = `{(8-6x; 0<x≤2), (4x-12; 2<x≤3),(2x+10; 3<x≤6):}` then f(x) is ______
If f(x) = `{{:(tanx/x + secx",", x ≠ 0),(2",", x = 0):}`, then ______.
If f(x) = `{{:(x, "for" x ≤ 0),(0,
"for" x > 0):}`, then f(x) at x = 0 is ______.
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.
If f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`
is continuous at x = 0, then the value of k is ______.
If \[\mathrm{f}(x)= \begin{cases} \mathrm{m}x+1, & x\leqslant\frac{\pi}{2} \\ \\ \mathrm{sin}x+\mathrm{n}, & x>\frac{\pi}{2} & \end{cases}\], is continuous at \[x=\frac{\pi}{2},( \begin{array} {c}\mathrm{m,n\in\mathbb{Z}} \end{array})\] then
Which is a necessary condition for continuity at \[x=c\]?
If \[f\] is defined on the closed interval \[[a,b]\], which condition establishes continuity at the left endpoint \[a\]?
If \[f\] is defined on the closed interval \[[a,b]\], which condition establishes continuity at the right endpoint \[b\]?
For \[f(x)=\begin{cases}1,&x\leq0\\2,&x>0\end{cases}\], why is \[f\] discontinuous at \[x=0\]?
