Advertisements
Advertisements
प्रश्न
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
विकल्प
e7
e3
e12
`"e"^(3/4)`
Advertisements
उत्तर
e12
Explanation;
f(x) is continuous at x = 0
∴ f(0) = `lim_(x -> 0) "f"(x)`
= `lim_(x -> 0) ((4 + 5x)/(4 - 7x))^(4/x)`
= `lim_(x -> 0) [(4(1 + (5x)/4))/(4(1 - (7x)/4))]^(4/x)`
= `(lim_(x -> 0) [(1 + (5x)/4)^(4/(5x))]^5)/(lim_(x -> 0)[(1 - (7x)/4)^((-4)/(7x))]^(-7))`
= `"e"^5/"e"^(-7) ...[(because x -> 0"," (5x)/4 -> 0"," (-7x)/4 -> 0),("and" lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= e12
APPEARS IN
संबंधित प्रश्न
Examine the continuity of f(x) = x3 + 2x2 − x − 2 at x = − 2
Examine the continuity of `"f"(x) {:(= sin x",", "for" x ≤ pi/4), (= cos x",", "for" x > pi/4):}} "at" x = pi/4`
Examine the continuity of `f(x) = {:((x^2 - 9)/(x - 3)",", "for" x ≠ 3),(=8",", "for" x = 3):}}` at x = 3.
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= ((27 - 2x)^(1/3) - 3)/(9 - 3(243 + 5x)^(1/5))",", "for" x ≠ 0),(= 2",", "for" x = 0):}}` at x = 0.
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) = `(x^2 - 10x + 21)/(x - 7)`
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`
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) `{:(= (x^3 - 8)/(x^2 - 4)",", "for" x > 2),(= 3",", "for" x = 2),(= ("e"^(3(x - 2)^2 - 1))/(2(x - 2)^2) ",", "for" x < 2):}`
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) = `(cos^2 x - sin^2 x - 1)/(sqrt(3x^2 + 1) - 1)` for x ≠ 0, is continuous at x = 0 then find f(0)
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
If f(x) `{:(= (5^x + 5^(-x) - 2)/(x^2)"," , "for" x ≠ 0),(= k",", "for" x = 0):}}` is continuous at x = 0, find k
For what values of a and b is the function
f(x) `{:(= (x^2 - 4)/(x - 2)",", "for" x < 2),(= "a"x^2 - "b"x + 3",", "for" 2 ≤ x < 3),(= 2x - "a" + "b"",", "for" x ≥ 3):}}` continuous for every x on R?
Determine the values of p and q such that the following function is continuous on the entire real number line.
f(x) `{:(= x + 1",", "for" 1 < x < 3),(= x^2 + "p"x + "q"",", "for" |x - 2| ≥ 1):}`
Show that there is a root for the equation 2x3 − x − 16 = 0 between 2 and 3.
Show that there is a root for the equation x3 − 3x = 0 between 1 and 2.
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) = `(x^2 - 7x + 10)/(x^2 + 2x - 8)`, for x ∈ [– 6, – 3]
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,
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:( = (sin^2pix)/(3(1 - x)^2) ",", "for" x ≠ 1),(= (pi^2sin^2((pix)/2))/(3 + 4cos^2 ((pix)/2)) ",", "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):}`
Find a and b if following function is continuous at the point or on the interval indicated against them:
f(x) `{:(= "a"x^2 + "b"x + 1",", "for" |2x - 3| ≥ 2),(= 3x + 2",", "for" 1/2 < x < 5/2):}`
Find f(a), if f is continuous at x = a where,
f(x) = `(1 + cos(pi x))/(pi(1 - x)^2)`, for x ≠ 1 and at a = 1
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) 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 ______
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 the function f(x) = `[tan(π/4 + x)]^(1/x)` for x ≠ 0 is = K for x = 0 continuous at x = 0, then K = ?
If f(x) = `{{:(x, "for" x ≤ 0),(0,
"for" x > 0):}`, then f(x) at x = 0 is ______.
If the function f(x) defined by
f(x) = `{{:(x sin 1/x",", "for" x = 0),(k",", "for" x = 0):}`
is continuous at x = 0, then k is equal to ______.
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 ?
Which of the following is not continuous for all x?
If f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`
is continuous at x = 0, then the value of k is ______.
If f(x) = `{{:((sin^3(sqrt(3)).log(1 + 3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3)) - 1)x)",", x ≠ 0),( a",", x = 0):}`
is continuous in [0, 1] then a is equal to ______.
`lim_(x rightarrow 0) (e^(x^2) - cosx)/x^2` is equal to ______.
Let `f(x) = (2 - sqrt(x + 4))/(sin 2x), x ≠ 0`. In order that f(x) is continuous at x = 0, f(0) is to be defined as ______.
