Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − 1)/x

Verified step by step guidance

1. Recall that the Taylor series expansion of the exponential function \(e^x\) around \(x=0\) is given by: \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]
2. Substitute the Taylor series expansion of \(e^x\) into the given limit expression: \[\frac{e^x - 1}{x} = \frac{\left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\right) - 1}{x}\]
3. Simplify the numerator by canceling the 1's: \[\frac{x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}{x}\]
4. Factor out \(x\) from the numerator: \[\frac{x \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \cdots \right)}{x}\]
5. Cancel the \(x\) terms and then evaluate the limit as \(x\) approaches 0 by substituting \(x=0\) into the remaining series inside the parentheses: \[\lim_{x \to 0} \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \cdots \right)\]

Key Concepts

Limits and Continuity

Limits describe the behavior of a function as the input approaches a particular value. Understanding limits is essential for analyzing functions near points where direct substitution may be undefined or indeterminate, such as 0/0 forms.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. Using Taylor series near zero (Maclaurin series) helps approximate functions and evaluate limits by simplifying complex expressions.

Indeterminate Forms and L'Hôpital's Rule

When evaluating limits, expressions like 0/0 are indeterminate forms that require special techniques. Taylor series or L'Hôpital's Rule can resolve these by transforming the expression into a determinate form to find the limit.