The value of _ x → 0 1 - cos^ 3 (sin x) sin xsin(sin x) (sin x) is equal to

Correct answer: A. (3)/(2) Let, sin x = t ⇒ _ t → 0 1 - cos^ 3 t tsin tcos t ⇒ _ t → 0 (1 - cos t)t^ 2 × ((t)/(sin t)) × (1 + cos t + ^ 2 t)cos t ⇒ (1)/(2) × 1 × ((1 + 1 + 1))/(1) = (3)/(2)

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Limits

NTA Abhyas 2020 · Mathematics · Limits

The value of _ x → 0 1 - cos^ 3 (sin x) sin xsin(sin x) (sin x) is...

The value of

\lim_{x \to 0}\dfrac{1 - cos^{3}\left(sin x\right)}{sin xsin\left(sin x\right)\cos \left(sin x\right)}

is equal to

Options

A $\dfrac{3}{2}$
B $1$
C $0$
D $2$

Correct answer

A. $\dfrac{3}{2}$

Step-by-step solution

Let,

sin x = t

⇒ \lim_{t \to 0}\dfrac{1 - cos^{3}t}{tsin tcos t}

⇒ \lim_{t \to 0}\left(1 - cos t\right)t^{2} \times \left(\dfrac{t}{sin t}\right) \times \left(1 + cos t + \cos^{2} t\right)cos t

⇒ \dfrac{1}{2} \times 1 \times \dfrac{\left(1 + 1 + 1\right)}{1} = \dfrac{3}{2}

Practice Limits on Quantrex Academy →

More from Limits

Let _ x → 2 (( (x-2))(rx^2 + (p-2)x - 2p))/((x-2)^2) = 5 for some r, p ∈ ℝ . If the set… JEE Main 2026 The value of _ x → 0 ((x^2 ^2 x)/(x^2 - ^2 x)) is: JEE Main 2026 Let f(x) = _ y → 0 ((1 - (xy)) (xy))/(y^3) . Then the number of solutions of the equation… JEE Main 2026 The product of all possible values of α , for which _ x → 0 ((1 - (α x) ((α+1)x)… JEE Main 2026 If _ x → 2 (x^3 - 5x^2 + ax + b) ( x-1 - 1) _e(x-1) = m , then a + b + m is equal to : JEE Main 2026 The value of _ x arrow 0 _ e ( (e x) · (e^ 2 x) · … · (e^ 10 x)) e^ 2 -e^ 2 x is equal to JEE Main 2026 If _ x arrow 0 e^ (a-1) x +2 ~b x+(c-2) e^ -x x x- _ e (1+x) =2 , then a^ 2 +b^ 2 +c^ 2… JEE Main 2026 Let [·] denote the greatest integer function and f(x)= _ n arrow ∞ 1 n^ 3 ∑_ k=1 ^ n [ k^… JEE Main 2026 Let f: R arrow(0, ∞) be a twice differentiable function such that f(3)=18, f^ (3)=0 and… JEE Main 2026 Let α and β be the real numbers such that _ x arrow 0 (1)/(x^3)((α)/(2) ∫_0^x (1)/(1-t^2)… JEE Advanced 2025 Let x_0 be the real number such that e^ x_0 +x_0=0 . For a given real number α , define… JEE Advanced 2025 Given below are two statements : Statement I : _ x arrow 0 ( ^ -1 x+ _e (1+x)/(1-x) -2 x… JEE Main 2025 If the function f(x)=( ( x)- ( x))/( x- x) is continuous at x=0 , then f(0) is equal to… JEE Main 2025 _ x arrow 0^ + (5(x)^ (1)/(3) ) _e(1+3 x^2) ( ^ -1 3 x )^2(e^ 5(x)^ (4)/(3) -1) is equal… JEE Main 2025 If _ x arrow 1^ + ((x-1)(6+λ (x-1))+μ (1-x))/((x-1)^3)=-1 , where λ, μ ∈ ℝ , then λ+μ is… JEE Main 2025 If Lim_ x arrow 0 (( x)/(x))^ (1)/(x^2) =p , then 96 _e p is equal to ______ JEE Main 2025 Full Limits list Limits JEE Maths PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs