Advertisements
Advertisements
Question
Find k if following function is continuous at the point indicated against them:
f(x) `{:(= ((5x - 8)/(8 - 3x))^(3/(2x - 4))",", "for" x ≠ 2),(= "k"",", "for" x = 2):}}` at x = 2
Advertisements
Solution
f(x) is continuous at x = 2
∴ f(2) = `lim_(x -> 2) "f"(x)`
∴ k = `lim_(x -> 2) ((5x - 8)/(8 - 3x))^(3/(2x - 4))`
Put x – 2 = h
∴ x = 2 + h
As x → 2, h → 0
∴ k = `lim_("h" -> 0) [(5(2 + "h") - 8)/(8 - 3(2 + "h"))]^(3/(2"h"))`
= `lim_("h" -> 0) ((10 + 5"h" - 8)/(8 - 6 - 3"h"))^(3/(2"h"))`
= `lim_("h" -> 0) ((2 + 5"h")/(2 - 3"h"))^(3/(2"h"))`
= `lim_("h" -> 0) [(2(1 + (5"h")/2))/(2(1 - (3"h")/2))]^(3/(2"h"))`
= `lim_("h" -> 0) (1 + (5"h")/2)^(3/(2"h"))/((1 - (3"h")/2)^(3/(2"h"))`
= `(lim_("h" -> 0) [(1 + (5"h")/2)^(2/(5"h"))]^(5/2 xx 3/2))/(lim_("h" -> 0)[(1 - (3"h")/2)^((-2)/(3"h"))]^((-3)/2 xx 3/2)`
= `"e"^(15/4)/"e"^((-9)/4) ...[(because "h" -> 0"," (5"h")/2 -> 0"," (-3"h")/2 -> 0),("and" lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^(24/4)`
= e6
APPEARS IN
RELATED QUESTIONS
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) = x3 + 2x2 − x − 2 at x = − 2
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) `{:(= (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) `{:(= 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 - 10x + 21)/(x - 7)`
Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :
f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, 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) `{:(= 3x + 2",", "for" -4 ≤ x ≤-2),(= 2x - 3";", "for" -2 < x ≤ 6):}`
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)
Show that there is a root for the equation 2x3 − x − 16 = 0 between 2 and 3.
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
Select the correct answer from the given alternatives:
f(x) `{:(= (32^x - 8^x - 4^x + 1)/(4^x - 2^(x + 1) + 1)",", "for" x ≠ 0),(= "k""," , "for" x = 0):}` is continuous at x = 0, then value of ‘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) `{:(= 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) `{:(= (|x + 1|)/(2x^2 + x - 1)",", "for" x ≠ -1),(= 0",", "for" x = -1):}}` at x = – 1
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) = [x + 1] for x ∈ [−2, 2)
Where [*] is greatest integer function.
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]):}`
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 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):}`
If f(x) = `{:{(tan^-1|x|; "when" x ≠ 0), (pi/4; "when" x = 0):}`, then ______
If f(x) = `[tan (pi/4 + x)]^(1/x)`, x ≠ 0 at
= k, x = 0 is continuous x = 0. Then k = ______.
If function `f(x)={((x^2-9)/(x-3), ",when "xne3),(k, ",when "x =3):}` is continuous at x = 3, then the value of k will be ______.
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) = `{{:((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 ______.
A real function \[f\] is a continuous function when it is continuous at which points?
Which condition makes a function discontinuous at \[x=a\]?
What is the graphical meaning of a function being continuous at a point?
Which statement about polynomial functions is correct?
For \[f(x)=x^2\], what verifies continuity at \[x=0\]?
For \[f(x)=|x|=\begin{cases}-x,&x<0\\x,&x\geq0\end{cases}\], what are the LHL, RHL, and \[f(0)\] at \[x=0\]?
