Limit Using Series Expansion

Use series to find the limit of $\frac{e^x - 1 - x}{x^2}$ as $x \to 0$.

To solve the given problem, one leverages the concept of series expansion, specifically Taylor and Maclaurin series, which are highly useful for finding limits and approximations near a specific point. The problem involves finding a limit as x approaches zero, a classic scenario for applying series expansions to simplify and solve the expression.

The Taylor series for a function provides an infinite sum of terms calculated from the values of its derivatives at a single point. In this instance, using the series expansion for the exponential function enables the simplification of the expression. The core idea is to expand e to the x as a series and then substitute this expansion into the given expression. By this approach, you will often cancel terms or group them into forms that make the limit straightforward.

Understanding the fundamentals of series, particularly Taylor and Maclaurin series, is critical in calculus, leading to elegant solutions for seemingly complex problems. It provides a powerful toolset for simplifying expressions and finding limits analytically, reinforcing concepts like differentiation and integration as you analyze the behavior of functions near specific points.