Advertisements
Advertisements
प्रश्न
x cos x
उत्तर
Let `y = x cos x` ......(i)
`y + Δy = (x + Δx) cos(x + Δx)` ......(ii)
Subtracting eq. (i) from equation (ii) we get
`y + Δy - y = (x + Δx) cos(x + Δx) - x cos x`
⇒ `Δy = x cos (x + Δx) + Δx cos (x + Δx) - x cos x`
Dividing both sides by Δx and take the limits,
`lim_(Δx -> 0) (Δy)/(Δx) = lim_(Δx -> 0) (x cos (x + Δx) - x cos x + Δx cos (x + Δx))/(Δx)`
`(dy)/(dx) = lim_(Δx -> 0) (x[cos(x + Δx) - cos x])/(Δx) + lim_(Δx -> 0) (Δx cos(x + Δx))/(Δx)` ......`["By defination" lim_(Δx -> 0) (Δy)/(Δx) = (dy)/(dx)]`
= `lim_(Δx -> 0) (x[-2 sin ((x + Δx + x))/2 * sin ((x + Δx - x))/2])/(Δx) + lim_(Δx -> 0) cos(x + Δx)`
= `lim_((Δx -> 0),(because (Δx)/2 -> 0)) (x[-2 sin(x + (Δx)/2) * sin (Δx)/2])/(2 xx (Δx)/2) + lim_(Δx - > 0) cos(x + Δx)`
∴ `(Δx)/2 -> 0` Taking the limits, we have
= `x[- sin x] + cos x` .......`[because lim_((Δx)/2 -> 0) (sin (Δx)/2)/((Δx)/2) = 1]`
= `- x sin x + cos x`
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit.
`lim_(x ->0) cos x/(pi - x)`
Evaluate the following limit :
`lim_(theta -> 0) [(1 - cos2theta)/theta^2]`
Evaluate the following limit :
`lim_(x -> 0)[(1 - cos("n"x))/(1 - cos("m"x))]`
Evaluate the following limit :
`lim_(x -> 0) [(cos("a"x) - cos("b"x))/(cos("c"x) - 1)]`
Select the correct answer from the given alternatives.
`lim_(x → π/3) ((tan^2x - 3)/(sec^3x - 8))` =
Evaluate the following :
`lim_(x -> 0) [(x(6^x - 3^x))/(cos (6x) - cos (4x))]`
Evaluate the following :
`lim_(x -> "a") [(x cos "a" - "a" cos x)/(x - "a")]`
`lim_{x→-5} (sin^-1(x + 5))/(x^2 + 5x)` is equal to ______
Evaluate `lim_(x -> pi/2) (secx - tanx)`
Evaluate `lim_(x -> pi/6) (2sin^2x + sin x - 1)/(2sin^2 x - 3sin x + 1)`
Evaluate `lim_(x -> 0) (cos ax - cos bx)/(cos cx - 1)`
`lim_(x -> pi/2) (1 - sin x)/cosx` is equal to ______.
`lim_(x -> 1) [x - 1]`, where [.] is greatest integer function, is equal to ______.
If f(x) = x sinx, then f" `pi/2` is equal to ______.
Evaluate: `lim_(x -> a) ((2 + x)^(5/2) - (a + 2)^(5/2))/(x - a)`
Evaluate: `lim_(x -> 3) (x^3 + 27)/(x^5 + 243)`
Evaluate: `lim_(x -> 0) (sin^2 2x)/(sin^2 4x)`
Evaluate: `lim_(x -> 0) (1 - cos mx)/(1 - cos nx)`
Evaluate: `lim_(x -> 0) (sin 2x + 3x)/(2x + tan 3x)`
Evaluate: `lim_(x -> pi/6) (cot^2 x - 3)/("cosec" x - 2)`
cos (x2 + 1)
`lim_(x -> 0) sinx/(sqrt(x + 1) - sqrt(1 - x)` is ______.
`lim_(x -> pi/4) (sec^2x - 2)/(tan x - 1)` is equal to ______.
If `lim_(x→∞) 1/(x + 1) tan((πx + 1)/(2x + 2)) = a/(π - b)(a, b ∈ N)`; then the value of a + b is ______.
If `lim_(n→∞)sum_(k = 2)^ncos^-1(1 + sqrt((k - 1)(k + 2)(k + 1)k)/(k(k + 1))) = π/λ`, then the value of λ is ______.
The value of `lim_(x rightarrow 0) (4^x - 1)^3/(sin x^2/4 log(1 + 3x))`, is ______.
`lim_(x rightarrow ∞) sum_(x = 1)^20 cos^(2n) (x - 10)` is equal to ______.
`lim_(x rightarrow π/2) ([1 - tan (x/2)] (1 - sin x))/([1 + tan (x/2)] (π - 2x)^3` is ______.
