Question

The value of $\underset{x \rightarrow 0}{l i m} \, \frac{1 - c o s^{3} \left(sin x\right)}{sin ⁡ x \, s i n \left(sin ⁡ x\right) c o s \left(sin ⁡ x\right)}$

is equal to

• A. $\frac{3}{2}$
• B. $1$
• C. $0$
• D. $2$

Answer 1

Answer: (A)

Let, $sin x=t$
$\Rightarrow \underset{t \rightarrow 0}{l i m}\frac{1 - c o s^{3} t}{t sin t cos ⁡ t}$
$\Rightarrow \underset{t \rightarrow 0}{l i m}\frac{\left(1 - cos t\right)}{t^{2}}\times \left(\frac{t}{sin ⁡ t}\right)\times \frac{\left(1 + cos ⁡ t + \left(cos\right)^{2} ⁡ t\right)}{cos ⁡ t}$
$\Rightarrow \frac{1}{2}\times 1\times \frac{\left(1 + 1 + 1\right)}{1}=\frac{3}{2}$