Use Maclaurin series to evaluate the limits. \(\lim _{x \rightarrow 0} \frac{1-e^{x^{3}}}{x^{3}}\)

In calculus, limits help us understand the behavior of a function as it approaches a specific point. When calculating limits, we're essentially zooming in on the function's value near that point. This concept is foundational for understanding continuous functions and forms the basis for derivatives and integrals.

Consider the limit we worked through: \(\[ \lim_{x \rightarrow 0} \frac{1 - e^{x^3}}{x^3} \]\). Understanding limits allows us to tackle expressions that might seem undefined at first glance.

The process often involves simplifying complex expressions, converting them using known mathematical series, and then observing their behavior as the variable approaches the specified value. It's crucial in solving many problems across various fields of math, physics, and engineering.

A series expansion is a way of expressing a function as a sum of terms calculated from the values of its derivatives at a single point. The Maclaurin series is a common type that expands a function around 0. It's a powerful tool to approximate functions using polynomials, making them easier to analyze and compute.

For example, the Maclaurin series for \(e^x\) is: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \] . By substituting \(x^3\) for \(x\), we transformed this series to: \[ e^{x^3} = 1 + x^3 + \frac{(x^3)^2}{2!} + \frac{(x^3)^3}{3!} + \cdots \] . This helps to break down complex expressions and find the solution step by step.

Using a series expansion to evaluate limits provides a clear path for simplification, as it converts the function into a polynomial form. Polynomials are much easier to handle mathematically than their original forms.

Evaluating limits involves several techniques, but using series expansion, like the Maclaurin series, is particularly useful for complex functions. By rewriting the function with a series, we turn it into a polynomial where the behavior near the point of interest (in this case, 0) is clearer.

Steps to evaluate \[ \lim_{x \rightarrow 0} \frac{1 - e^{x^3}}{x^3} \] using Maclaurin series:

1. **Recall the Maclaurin Series**: Start with \( e^x \) and write its series.
2. **Substitute \(x^3\) into the series**: Transform \(e^x\) to \(e^{x^3}\).
3. **Simplify and Rearrange**: Subtract and divide to match the given limit expression.
4. **Observe the behavior as \(x \rightarrow 0\)**: Notice how higher-order terms vanish, making it straightforward to find the limit.

This method helps confirm our result intuitively and analytically. It makes complex calculations manageable and offers insight into the function's behavior around specific points.