Advertisements
Advertisements
प्रश्न
If f(x) = `(4^(x - π) + 4^(π - x) - 2)/(x - π)^2` for x ≠ π, is continuous at x = π, then find f(π).
Advertisements
उत्तर
f is given to be continuous at x = π
∴ by definition,
f(π) = `lim_(x -> pi) "f"(x)`
= `lim_(x -> pi) (4^(x - pi) + 4^(pi - x) - 2)/(x - pi)^2`
Put x = π + h. Then as x → π, h → 0 and x – π = h
∴ f(π) = `lim_("h" -> 0) (4^"h" + 4^(-"h") - 2)/"h"^2`
= `lim_("h" -> 0) (4^"h"[4^"h" + 4^(-"h") - 2])/(4^"h" * "h"^2)`
= `lim_("h" -> 0) ((4^"h")^2 + 1 - 2(4^"h"))/(4^"h" * "h"^2)`
= `lim_("h" -> 0) (4^"h" - 1)^2/(4^"h" * "h"^2)`
= `[lim_("h" -> 0) (4^"h" - 1)/"h"]^2/(lim_("h" -> 0) 4^"h")`
= `(log 4)^2/4^0 ...[lim_(x -> 0) ("a"^x - 1)/x = log "a"]`
= (2 log 2)2
∴ f(π) = 4(log 2)2
APPEARS IN
संबंधित प्रश्न
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 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`
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= 4x + 1",", "for" x ≤ 8/3),(= (59 - 9x)/3 ",", "for" x > 8/3):}} "at" x = 8/3`
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) `{:(= x^2 - 3x - 2",", "for" x < -3),(= 3 + 8x",", "for" x > -3):}`
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) = `{:(=( sqrt(3) - tanx)/(pi - 3x)",", x ≠ pi/3),(= 3/4",", x = pi/3):}} "at" x = pi/3`
Discuss the continuity of the following function at the point indicated against them :
f(x) `{:(=("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "for" x ≠ 0),(= 1",", "for" x = 0):}}` at 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 becomes continuous :
f(x) `{:(=("e"^(5sinx) - "e"^(2x))/(5tanx - 3x)",", "for" x ≠ 0),(= 3/4",", "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) `{:(= 3x + 2",", "for" -4 ≤ x ≤-2),(= 2x - 3";", "for" -2 < x ≤ 6):}`
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)`
If f(x) `{:(= (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)",", "for" x ≠ 0), (= "k"",", "for" x = 0):}}` is continuous at x = 0, find k
Show that there is a root for the equation 2x3 − x − 16 = 0 between 2 and 3.
Select the correct answer from the given alternatives:
If f(x) = `((sin2x)tan5x)/("e"^(2x) - 1)^2`, for x ≠ 0 is continuous at x = 0, then f(0) is
Select the correct answer from the given alternatives:
f(x) `{:(= ((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))",", "for" x ≠ 0),(= "k"",", "for" x = 0):}` is continuous at x = 0, then ‘k’ =
Select the correct answer from the given alternatives:
If f(x) = `((4 + 5x)/(4 - 7x))^(4/x)`, for x ≠ 0 and f(0) = k, is continuous at x = 0, then k is
Select the correct answer from the given alternatives:
If f(x) = [x] for x ∈ (–1, 2) then f is discontinuous at
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`
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= (|x + 1|)/(2x^2 + x - 1)",", "for" x ≠ -1),(= 0",", "for" x = -1):}}` at x = – 1
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= x^2 + 5x + 1"," , "for" 0 ≤ x ≤ 3),(= x^3 + x + 5"," , "for" 3 < x ≤ 6):}`
Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:
f(x) = `((x + 3)(x^2 - 6x + 8))/(x^2 - x - 12)`
Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:
f(x) `{:(= x^2 + 2x + 5"," , "for" x ≤ 3),( = x^3 - 2x^2 - 5",", "for" x > 3):}`
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.
Let f : [-1, 2] → [0, ∞] be a continuous function such that f(x) = f(1 - x) ∀ x ∈ [-1, 2].
Let R1 = `int_-1^2 xf(x) dx` and R2 be the area of the region bounded by y = f(x), x = -1, x = 2 and the X-axis. Then, ______
If f(x) `= sqrt(4 + "x" - 2)/"x", "x" ne 0` be continuous at x = 0, then f(0) = ______.
Let f be the function defined by
f(x) = `{{:((x^2 - 1)/(x^2 - 2|x - 1| - 1)",", x ≠ 1),(1/2",", x = 1):}`
If f(x) = `{{:(x, "for" x ≤ 0),(0,
"for" x > 0):}`, then f(x) at x = 0 is ______.
For what value of k, the function defined by
f(x) = `{{:((log(1 + 2x)sin^0)/x^2",", "for" x ≠ 0),(k",", "for" x = 0):}`
is continuous at x = 0 ?
If f(x) = `{{:((x - 4)/(|x - 4|) + a",", "for" x < 4),(a + b",", "for" x = 4 "is continuous at" x = 4","),((x - 4)/(|x - 4|) + b",", "for" x > 4):}`
then ______.
