From the uploaded image, it seems you're working with a mathematical limit expression, potentially after applying a Taylor series expansion to simplify the problem. Let me break this down step by step for you.

The primary expressions visible in the image include:

1. Taylor Expansion Result:

   • In the numerator: $\left( -\frac{x^2}{2} + S(x) \right)^3$
   • In the denominator: $\left( -\frac{x^3}{6} + R_3(x) \right)^2$

2. Limit Form:

   • $\lim_{x \to \infty} \frac{\text{Numerator}}{\text{Denominator}}$

I will simplify this mathematically step by step, based on your Taylor-based derivation.

Step 1: Simplify the Taylor Terms

Numerator:

$\left( -\frac{x^2}{2} + S(x) \right)^3$ Here, $S(x)$ is presumably a smaller-order term (e.g., $O(x)$, $O(1)$, or an exponential decay). For large $x$, the dominant term in the numerator is: $\left( -\frac{x^2}{2} \right)^3 = -\frac{x^6}{8}.$

Denominator:

$\left( -\frac{x^3}{6} + R_3(x) \right)^2$ Similarly, $R_3(x)$ is likely a smaller-order term. For large $x$, the dominant term in the denominator is: $\left( -\frac{x^3}{6} \right)^2 = \frac{x^6}{36}.$

Step 2: Form the Limit

Using the dominant terms from both numerator and denominator: $\lim_{x \to \infty} \frac{\left( -\frac{x^2}{2} \right)^3}{\left( -\frac{x^3}{6} \right)^2}$

Substitute the dominant terms: $\lim_{x \to \infty} \frac{-\frac{x^6}{8}}{\frac{x^6}{36}}$

Step 3: Simplify the Fraction

$\frac{-\frac{x^6}{8}}{\frac{x^6}{36}} = -\frac{\frac{1}{8}}{\frac{1}{36}} = -\frac{36}{8} = -4.5$

Final Result:

$\lim_{x \to \infty} \frac{\left( -\frac{x^2}{2} + S(x) \right)^3}{\left( -\frac{x^3}{6} + R_3(x) \right)^2} = -4.5$

Let me know if you'd like further clarification or a deeper dive into any step!

Related Questions:

1. What is a Taylor series, and how is it used in limits?
2. How do dominant terms affect the simplification of large-limit problems?
3. Why do smaller-order terms become negligible as $x \to \infty$?
4. Can this method be extended to other types of functions, such as trigonometric or exponential ones?
5. What is the role of $O(x)$-notation in such expansions?

Tip:

When applying Taylor series, always confirm the order of approximation needed to ensure accuracy in limits. Keep track of higher-order terms when they might significantly affect the result.