Advertisements
Advertisements
प्रश्न
Find a and b if following function is continuous at the point or on the interval indicated against them:
f(x) `{:(= (4tanx + 5sinx)/("a"^x - 1)",", "for" x < 0),(= (9)/(log2)",", "for" x = 0),(= (11x + 7x*cosx)/("b"^x - 1)",", "for" x > 0):}`
Advertisements
उत्तर
f(x) is continuous at x = 0
∴ `lim_(x -> 0^-) "f"(x)` = f(0)
∴ `lim_(x -> 0) [(4tanx + 5sinx)/("a"^x - 1)] = 9/log2`
∴ `lim_(x -> 0) [((4tanx + 5sinx)/x)/(("a"^x - 1)/x)]` ...[∵ x → 0, x ≠ 0]
= `9/log2`
∴ `(lim_(x -> 0)((4tanx)/x + (5sinx)/x))/(lim_(x -> 0) ("a"^x - 1)/x) = 9/log2`
∴ `(4lim_(x -> 0) (tanx)/x + 5 lim_(x -> 0) (sinx)/x)/(lim_(x -> 0) ("a"^x - 1)/x) = 9/log2`
∴ `(4(1) + 5(1))/(log"a") = 9/log2 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
∴ `9/log"a" = 9/log2`
∴ log a = log 2
∴ a = 2
Also `lim_(x -> 0^+) "f"(x)` = f(0)
∴ `lim_(x -> 0) (11x + 7x*cosx)/("b"^x - 1) = 9/log2`
∴ `lim_(x -> 0) ((11x + 7x cosx)/x)/(("b"^x - 1)/x) = 9/log2` ...[∵ x → 0, x ≠ 0]
∴ `(lim_(x -> 0)(11 + 7cosx))/(lim_(x -> 0)(("b"^x - 1)/x)) = 9/log2`
∴ `(11 + 7cos0)/log"b" = 9/log2 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
∴ `(11 + 7(1))/log"b" = 9/log2`
∴ 9log b = 18log 2
∴ log b = 2log 2
= log(2)2
∴ log b = log 4
∴ b = 4
∴ a = 2 and b = 4
APPEARS IN
संबंधित प्रश्न
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) `{:(= (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)",", "for" x ≠ 2),(= 1/5",", "for" x = 2):}}`at x = 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) `{:(= 4x + 1",", "for" x ≤ 8/3),(= (59 - 9x)/3 ",", "for" x > 8/3):}} "at" x = 8/3`
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`
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) `{:(= log_((1 + 3x)) (1 + 5x)",", "for" x > 0),(=(32^x - 1)/(8^x - 1)",", "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):}`
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.
Let f(x) = ax + b (where a and b are unknown)
= x2 + 5 for x ∈ R
Find the values of a and b, so that f(x) is continuous at x = 1
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:
If f(x) = `(1 - sqrt(2) sinx)/(pi - 4x), "for" x ≠ pi/4` is continuous at x = `pi/4`, then `"f"(pi/4)` =
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
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 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
Solve using intermediate value theorem:
Show that x3 − 5x2 + 3x + 6 = 0 has at least two real root between x = 1 and x = 5
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) = `{(8-6x; 0<x≤2), (4x-12; 2<x≤3),(2x+10; 3<x≤6):}` then f(x) is ______
If f(x) `= sqrt(4 + "x" - 2)/"x", "x" ne 0` be continuous at x = 0, then f(0) = ______.
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 ?
The function f(x) = x – |x – x2| is ______.
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` 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 ______.
A real function \[f\] is a continuous function when it is continuous at which points?
When is \[f(x)\] non-removable discontinuous at \[x=a\]?
What is the graphical meaning of a function being continuous at a point?
For \[f(x)=\begin{cases}1,&x\leq0\\2,&x>0\end{cases}\], why is \[f\] discontinuous at \[x=0\]?
For which values is the constant function \[f(x)=k\] continuous?
For which values is the identity function \[f(x)=x\] continuous?
For a piecewise function, at which points should continuity be checked especially?
What does the greatest integer function \[f(x)=[x]\] give?
For \[f(x)=x^2\], what verifies continuity at \[x=0\]?
