Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
lim ₓ→₀ (eᵃˣ − 1)/x

Verified step by step guidance

1. Identify the limit expression: \(\lim_{x \to 0} \frac{e^{a x} - 1}{x}\), where \(a\) is a nonzero real parameter.
2. Recall the Taylor series expansion of the exponential function around \(x=0\): \(e^{a x} = 1 + a x + \frac{(a x)^2}{2!} + \frac{(a x)^3}{3!} + \cdots\).
3. Substitute the Taylor series expansion into the limit expression: \(\frac{e^{a x} - 1}{x} = \frac{\left(1 + a x + \frac{(a x)^2}{2!} + \cdots \right) - 1}{x} = \frac{a x + \frac{(a x)^2}{2!} + \cdots}{x}\).
4. Simplify the fraction by dividing each term in the numerator by \(x\): \(\frac{a x}{x} + \frac{(a x)^2}{2! x} + \cdots = a + \frac{a^2 x}{2!} + \cdots\).
5. Evaluate the limit as \(x \to 0\): since all terms involving \(x\) vanish, the limit simplifies to \(a\).

Key Concepts

Limits and Continuity

Limits describe the behavior of a function as the input approaches a particular value. Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms like 0/0, is essential for analyzing function behavior near specific points.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. Using the Taylor series around zero (Maclaurin series) helps approximate functions near that point, simplifying limit evaluation by replacing complex expressions with polynomial forms.

Exponential Function and Parameters

The exponential function e^(ax) depends on the parameter 'a' and variable 'x'. Understanding how the parameter affects the function's behavior and its derivatives is crucial when expressing limits in terms of 'a', especially when using series expansions to evaluate limits involving parameters.