Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (tan ⁻¹ x − x)/x³"

Verified step by step guidance

1. Recognize that the problem asks to evaluate the limit \( \lim_{x \to 0} \frac{\tan^{-1} x - x}{x^3} \) using Taylor series expansions.
2. Recall the Taylor series expansion of \( \tan^{-1} x \) around \( x = 0 \): \[ \tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots \]
3. Substitute the Taylor series expansion into the numerator: \[ \tan^{-1} x - x = \left(x - \frac{x^3}{3} + \cdots \right) - x = - \frac{x^3}{3} + \cdots \]
4. Rewrite the original limit expression using this substitution: \[ \lim_{x \to 0} \frac{\tan^{-1} x - x}{x^3} = \lim_{x \to 0} \frac{- \frac{x^3}{3} + \cdots}{x^3} \]
5. Simplify the fraction by dividing each term by \( x^3 \), then evaluate the limit by letting \( x \to 0 \), which will eliminate higher order terms, leaving the coefficient of the leading term.

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 limits, Taylor expansions help identify dominant terms and simplify evaluation.

Inverse Tangent Function (arctan) Properties

The inverse tangent function, arctan(x), is smooth and differentiable around zero, with a known Taylor series expansion. Understanding its series helps express arctan(x) as x minus higher-order terms, which is essential for evaluating limits involving arctan(x) near zero.

Limit Evaluation Using Series Expansion

When direct substitution in a limit leads to an indeterminate form, expanding functions into their Taylor series can reveal the behavior of the numerator and denominator. By comparing the lowest-order nonzero terms, one can compute the limit accurately without complex algebraic manipulation.