Advertisements
Advertisements
Question
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’ =
Options
`8/3`
`8/15`
`-8/15`
`20/3`
Advertisements
Solution
`8/15`
Explanation:
f(x) is continuous at x = 0
`therefore "f"(0) = lim_(x -> 0) (x)`
`therefore "k" = lim_(x -> 0)((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))`
`= (lim_(x -> 0) ((16^x - 1)/x) xx lim_(x -> 0) ((9^x - 1)/x))/(lim_(x -> 0) ((27^x - 1)/x) xx lim_(x -> 0) ((32^x - 1)/x))`
`= (log 16 xx log 9)/(log 27 xx log 32) ...[because lim_(x -> 0) ("a"^x - 1)/x = log "a"]`
`= (4 log 2 xx 2 log 3)/(3 log 3 xx 5 log 2)`
`= (4 cancel(log 2) xx 2 cancel(log 3))/(3 cancel(log 3) xx 5 cancel(log 2))`
`= (4 xx 2)/(3 xx 5)`
`= 8/15`
APPEARS IN
RELATED QUESTIONS
Examine whether the function is continuous at the points indicated against them:
f(x) = x3 − 2x + 1, for x ≤ 2
= 3x − 2, for x > 2, at x = 2
Examine the continuity of `f(x) = {:((x^2 - 9)/(x - 3)",", "for" x ≠ 3),(=8",", "for" x = 3):}}` at x = 3.
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
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`
Test the continuity of the following function at the point or interval indicated against them:
f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",", "for" 0 < x < 3"," x ≠ 2),(= 12",", "for" x = 2),(= (2 - 2x - x^2)/(x - 4)",", "for" 3 ≤ x < 4):}}` at x = 2
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) = `(x^2 - 10x + 21)/(x - 7)`
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) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for 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) `{:(= 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):}`
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 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:
f(x) = `{:(= (2^(cotx) - 1)/(pi - 2x)",", "for" x ≠ pi/2),(= log sqrt(2)",", "for" x = pi/2):}`
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) `{:(= 2x^2 + x + 1",", "for" |x - 3| ≥ 2),(= x^2 + 3",", "for" 1 < x < 5):}`
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]):}`
Find k if following function is continuous at the point indicated against them:
f(x) `{:(= (45^x - 9^x - 5^x + 1)/(("k"^x - 1)(3^x - 1))",", "for" x ≠ 0),(= 2/3",", "for" x = 0):}}` at 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):}`
Solve using intermediate value theorem:
Show that 5x − 6x = 0 has a root in [1, 2]
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) = `{:{(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 f(x) = `{{:(tanx/x + secx",", x ≠ 0),(2",", x = 0):}`, then ______.
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) = `{{:(log(sec^2 x)^(cot^2x)",", "for" x ≠ 0),(K",", "for" x = 0):}`
is continuous at x = 0, then K 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 x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.
If f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`
is continuous at x = 0, then the value of k is ______.
`lim_(x rightarrow 0) (e^(x^2) - cosx)/x^2` is equal to ______.
