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

Verified step by step guidance

1. Recall the Taylor series expansions of the exponential functions around 0: \(e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\) and \(e^{-x} = 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots\).
2. Substitute these expansions into the expression \(\frac{e^{x} - e^{-x}}{x}\) to get \(\frac{\left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\right) - \left(1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots\right)}{x}\).
3. Simplify the numerator by combining like terms: the constant terms cancel out, and you combine the \(x\) terms, \(x^{2}\) terms, etc., carefully noting the signs.
4. After simplification, factor out \(x\) from the numerator to cancel with the denominator, which will help in evaluating the limit as \(x\) approaches 0.
5. Evaluate the resulting expression by substituting \(x = 0\) into the simplified series to find the limit.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It approximates functions near that point, allowing complex expressions to be simplified into polynomials. For example, eˣ can be expanded around x=0 as 1 + x + x²/2! + x³/3! + ....

Limit Evaluation Using Series

When direct substitution in a limit leads to an indeterminate form, expanding functions into their Taylor series can simplify the expression. By substituting series expansions, terms can often be canceled or simplified, making it easier to find the limit as x approaches a point.

Hyperbolic Sine Function and Its Relation

The expression (eˣ − e⁻ˣ)/2 defines the hyperbolic sine function, sinh(x). Recognizing this helps simplify the limit problem since (eˣ − e⁻ˣ)/x = 2 * (sinh(x)/x). Understanding sinh(x) and its series expansion aids in evaluating limits involving exponential differences.