Limits by power series Use Taylor series to evaluate the following limits.


lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

Verified step by step guidance

1. Recall the Taylor series expansion of \( \cos x \) around \( x = 0 \): \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]
2. Substitute the Taylor series expansion of \( \cos x \) into the given expression: \[ \frac{\frac{x^2}{2} - 1 + \cos x}{x^4} = \frac{\frac{x^2}{2} - 1 + \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \right)}{x^4} \]
3. Simplify the numerator by combining like terms: \[ \frac{x^2}{2} - 1 + 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots = \frac{x^4}{24} - \cdots \]
4. Rewrite the expression with the simplified numerator: \[ \frac{\frac{x^4}{24} - \cdots}{x^4} \]
5. Divide each term in the numerator by \( x^4 \) and then take the limit as \( x \to 0 \). The higher order terms vanish, leaving the constant term from the division.

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 like cosine to be expressed as polynomials, which simplifies limit evaluation.

Limit Evaluation Using Series

By substituting the Taylor series into the limit expression, we can rewrite the limit in terms of powers of x. This method helps identify dominant terms and simplifies the limit calculation, especially when direct substitution leads to indeterminate forms.

Handling Indeterminate Forms

Limits that result in forms like 0/0 require algebraic manipulation or series expansion to resolve. Using power series expansions transforms the expression into a form where the limit can be directly computed by canceling terms or evaluating the leading coefficients.