Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Find all the point of discontinuities of f(x) = [x] on the interval (− 3, 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) `{:(= (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`
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):}`
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
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) = `{:(=( 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 become continuous :
f(x) `{:(= 3x + 2",", "for" -4 ≤ x ≤-2),(= 2x - 3";", "for" -2 < x ≤ 6):}`
For what values of a and b is the function
f(x) `{:(= "a"x + 2"b" + 18",", "for" x ≤ 0),(= x^2 + 3"a" - "b"",", "for" 0 < x ≤ 2),(= 8x - 2",", "for" x > 2):}}` continuous for every x?
Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for" -3 ≤ x ≤ 2),(= |x - 5|",", "for" 2 < x ≤ 7):}`.
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):}`
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) = `(x^2 - 7x + 10)/(x^2 + 2x - 8)`, for x ∈ [– 6, – 3]
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
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.
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
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):}`
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[7(x - pi)])/(5(x - pi)^2`, for x ≠ π at a = π
If f(x) = `{:{(tan^-1|x|; "when" x ≠ 0), (pi/4; "when" x = 0):}`, then ______
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) = `{{:((sin5x)/(x^2 + 2x)",", x ≠ 0),(k + 1/2",", x = 0):}` is continuous at x = 0, then the value of k is ______.
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 ______.
Which condition defines continuity of a real-valued function \[f\] at \[x=c\]?
If \[f\] is defined on the closed interval \[[a,b]\], which condition establishes continuity at the right endpoint \[b\]?
Which condition makes a function discontinuous at \[x=a\]?
When is \[f(x)\] removable discontinuous at \[x=a\]?
For \[f(x)=\begin{cases}1,&x\leq0\\2,&x>0\end{cases}\], why is \[f\] discontinuous at \[x=0\]?
For a piecewise function, at which points should continuity be checked especially?
