Use series to evaluate the limits in Exercises 29–40.
37. lim (x → 0) ln(1 + x²) / (1 - cos(x))

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Recall the Maclaurin series expansions for the functions involved: \( \ln(1 + x^2) \) and \( \cos(x) \).

Write the series expansion for \( \ln(1 + x^2) \) around \( x = 0 \). Since \( \ln(1 + u) = u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \), substitute \( u = x^2 \) to get \( \ln(1 + x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \cdots \).

Write the series expansion for \( \cos(x) \) around \( x = 0 \): \( \cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \).

Substitute the series expansions into the original limit expression: \( \frac{\ln(1 + x^2)}{1 - \cos(x)} = \frac{x^2 - \frac{x^4}{2} + \cdots}{1 - \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \right)} \).

Simplify the denominator: \( 1 - \cos(x) = \frac{x^2}{2} - \frac{x^4}{24} + \cdots \). Then, divide the numerator series by the denominator series, and identify the leading terms to find the limit as \( x \to 0 \).

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Taylor and Maclaurin Series

Taylor and Maclaurin series express functions as infinite sums of powers of (x - a) or x, respectively. They allow approximation of functions near a point, making it easier to analyze limits, derivatives, and integrals. For example, ln(1 + x²) and cos(x) can be expanded into power series around x = 0.

Limit Evaluation Using Series Expansion

When direct substitution in a limit leads to an indeterminate form, series expansions help by approximating numerator and denominator with polynomials. This simplification often reveals the limit's value by canceling terms or comparing leading powers, especially useful for limits involving logarithmic and trigonometric functions.

Behavior of Logarithmic and Trigonometric Functions Near Zero

Understanding how ln(1 + x²) and cos(x) behave near zero is crucial. ln(1 + x²) behaves like x² for small x, while cos(x) approximates 1 - x²/2. Recognizing these approximations helps simplify the limit expression and identify dominant terms in numerator and denominator.

Use series to evaluate the limits in Exercises 29–40.37. lim (x → 0) ln(1 + x²) / (1 - cos(x))