Advertisements
Advertisements
Question
Evaluate the following :
`lim_(x -> 0) [(x(6^x - 3^x))/(cos (6x) - cos (4x))]`
Advertisements
Solution
`lim_(x -> 0) (x(6^x - 3^x))/(cos (6x) - cos (4x))`
= `lim_(x -> 0) (x*3^x (2^x - 1))/(-2sin ((6x + 4x)/2) sin((6x - 4x)/2))`
= `1/(-2) lim_(x -> 0) (x*3^x (2^x - 1))/(sin 5x*sinx)`
= `1/(-2) lim_(x -> 0) ((x*3^x (2^x - 1))/x^2)/((sin 5x* sinx)/x^2) ...[("Divide Numerator and"),("Denominator by" x^2),(∵ x -> 0"," ∴ x ≠ 0 ∴ x^2 ≠ 0)]`
= `1/(-2) (lim_(x -> 0) [3^x ((2^x - 1))/x])/(lim_(x -> 0) ((sin5x)/x * sinx/x))`
= `1/(-2) (lim_(x -> 0)(3x) * lim_(x -> 0) ((2^x - 1)/x))/(lim_(x -> 0) (sinx/(5x) xx 5) * lim_(x -> 0) (sinx/x))`
= `1/(-2) xx (3^circ xx log 2)/(1 xx 5 xx 1) ...[(because x -> 0"," 5x -> 0),(lim_(x -> 0) ("a"^x - 1)/x = log "a"),(lim_(x -> 0) sinx/x = 1)]`
= `(-1)/10 log 2`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit.
`lim_(x -> (pi)/2) (tan 2x)/(x - pi/2)`
Evaluate the following limit :
`lim_(theta -> 0) [(sin("m"theta))/(tan("n"theta))]`
Evaluate the following limit :
`lim_(x -> 0) [(x*tanx)/(1 - cosx)]`
Evaluate the following limit :
`lim_(x -> pi/4) [(cosx - sinx)/(cos2x)]`
Evaluate the following limit :
`lim_(x -> 0) [(cos("a"x) - cos("b"x))/(cos("c"x) - 1)]`
Evaluate the following :
`lim_(x -> "a") [(sinx - sin"a")/(x - "a")]`
Evaluate the following :
`lim_(x -> "a") [(x cos "a" - "a" cos x)/(x - "a")]`
`lim_{x→0}((3^x - 3^xcosx + cosx - 1)/(x^3))` is equal to ______
Evaluate `lim_(x -> 2) 1/(x - 2) - (2(2x - 3))/(x^3 - 3x^2 + 2x)`
Evaluate `lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x))`
Evaluate: `lim_(x -> 1/2) (4x^2 - 1)/(2x - 1)`
Evaluate: `lim_(x -> 0) ((x + 2)^(1/3) - 2^(1/3))/x`
Evaluate: `lim_(x -> sqrt(2)) (x^4 - 4)/(x^2 + 3sqrt(2x) - 8)`
Evaluate: `lim_(x -> 1) (x^7 - 2x^5 + 1)/(x^3 - 3x^2 + 2)`
Evaluate: `lim_(x -> 3) (x^3 + 27)/(x^5 + 243)`
Evaluate: `lim_(x -> 0) (1 - cos mx)/(1 - cos nx)`
Evaluate: `lim_(x -> 0) (sin 2x + 3x)/(2x + tan 3x)`
Evaluate: `lim_(x -> a) (sin x - sin a)/(sqrt(x) - sqrt(a))`
Evaluate: `lim_(x -> pi/6) (cot^2 x - 3)/("cosec" x - 2)`
Evaluate: `lim_(x -> 0) (sqrt(2) - sqrt(1 + cos x))/(sin^2x)`
`(ax + b)/(cx + d)`
x cos x
`lim_(x -> pi) (1 - sin x/2)/(cos x/2 (cos x/4 - sin x/4))`
Show that `lim_(x -> 4) |x - 4|/(x - 4)` does not exists
`lim_(x -> pi) sinx/(x - pi)` is equal to ______.
`lim_(x -> 0) (x^2 cosx)/(1 - cosx)` is ______.
`lim_(x -> 0) ("cosec" x - cot x)/x` is equal to ______.
`lim_(x -> pi/4) (sec^2x - 2)/(tan x - 1)` is equal to ______.
`lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(2x^2 + x - 3)` is ______.
If `f(x) = {{:(x^2 - 1",", 0 < x < 2),(2x + 3",", 2 ≤ x < 3):}`, the quadratic equation whose roots are `lim_(x -> 2^-) f(x)` and `lim_(x -> 2^+) f(x)` is ______.
`lim_(x rightarrow ∞) sum_(x = 1)^20 cos^(2n) (x - 10)` is equal to ______.
`lim_(theta → -pi/4) (cos theta + sin theta)/(theta + pi/4)` =
