The value of $\lim_{x \to 0} \frac{\sin\left(\frac{x}{3}\right) \sin\left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}$ is equal to
Step-by-Step Solution
Key Concept: When dealing with limits of ratios of similar expressions, dividing by the dominant term and using functional properties simplifies the computation.
Rewrite the limit as $\lim_{x \to 0} \frac{u(x^2) + ax^4}{u(x^2) + ax^4} = \lim_{x \to 0} \frac{u(x^2) + ax^4}{u(x^2) + bx^4}$. Dividing numerator and denominator by $u(x^2)$ and using $\frac{u(x^2)}{x^4} \to L$, we get $\lim_{x \to 0} \frac{1 + aL}{1 + bL} = \frac{1600 + 5}{200 + 5} = \frac{1605}{205} = \frac{5}{1}$. Therefore the answer is $5$.
Correct Answer: 5