Question

$\lim_{x\rightarrow0}\frac{x \cos^{2}x}{\sin x}$ is equal to ________.

• A. 4
• B. 2
• C. -2
• D. 0
• E. 1

Hint:

$\lim_{x\to 0} \frac{x}{\sin x}$ is 1.

Answer 1

Answer: (E)

Step 1: Concept
Rearrange the limit to use the standard identity $\lim_{x\to 0} \frac{\sin x}{x} = 1$.

Step 2: Meaning
The expression can be written as $(\frac{x}{\sin x}) \cdot \cos^2 x$.

Step 3: Analysis
As $x \to 0$, $\frac{x}{\sin x} \to 1$ and $\cos^2(0) \to 1^2 = 1$.

Step 4: Conclusion
The product of the limits is $1 \times 1 = 1$. Final Answer: (E)