Question

Evaluate: $\underset{x\to 0}{\lim }\frac{d}{dx}\left[\int \left(\frac{(1-\cos x)}{x^2}\right)dx\right]$

• A. $1$
• B. $0$
• C. $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is C

$\frac{1}{2}$

Explanation for the correct answer:

Simplifying the equation to determinate form and applying the limits:

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\frac{d}{dx}\left[\int \left(\frac{(1-\cos x)}{x^2}\right)dx\right] \\ \Rightarrow \underset{x\to 0}{\lim }\left(\frac{(1-\cos x)}{x^2}\right)[\because \text{Differntiation and integration gets cancelled}] \\ \Rightarrow \underset{x\to 0}{\lim }\left(\frac{-(-\sin x)}{2x}\right)\left[\because \frac{d}{dx}\frac{(1-\cos x)}{x^2}=\frac{-(-\sin x)}{2x}\right] \end{gathered}$$

Applying the limits

$$\begin{gathered} \Rightarrow \frac{1}{2}\times 1\left[\because \underset{x\to 0}{\lim }\frac{\sin \theta }{\theta }=1\right] \\ \Rightarrow \frac{1}{2} \end{gathered}$$

Thus, $\underset{x\to 0}{\lim }\frac{d}{dx}\left[\int \left(\frac{(1-\cos x)}{x^2}\right)dx\right]=\frac{1}{2}$

Therefore, the correct answer is option (C).