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The Taylor series expansion is a powerful tool in calculus used to approximate functions using polynomials. It involves expressing a function as an infinite sum of terms calculated from the function's derivatives at a point. For instance, the expansion for \( \cos(x) \) around zero (Maclaurin series) is: \[\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots\]This approximation becomes very useful for calculating limits, especially when the function involves trigonometric, exponential, or logarithmic components.
Another example is the expansion of \(\log(1 + x^2)\) around zero:\[\log(1 + x^2) = x^2 - \frac{x^4}{2} + O(x^6)\]The series provides polynomial approximations to functions that can simplify the process of evaluating limits when \( x \) is close to zero. Just remember, the accuracy of the approximation increases with the number of terms included from the series.

Limits are fundamental in calculus and are used to describe the behavior of functions as inputs approach a specific value. In the context of Taylor Series, limits help simplify complex expressions by approximating them using polynomials.
The limit \( \lim_{x \to 0} \frac{1 - \cos (x)}{x^{2}} \) is evaluated using the Taylor expansion of \( \cos(x) \). Substituting the expansion into the limit expression and simplifying step by step allows us to find the result.
Similarly, evaluating \( \lim_{x \to 0} \frac{\log (1 + x^{2})}{2 x} \) involves substituting the Taylor series of the logarithmic function and simplifying. Understanding these steps is crucial because they demonstrate how applying polynomial approximations can make finding the limit more straightforward.

Numerical analysis involves techniques for approximating mathematical procedures. When it comes to analyzing functions, Taylor polynomials provide a bridge between exact values and numerical approximations.
By using the Taylor series' several terms, we can understand the function's behavior near a specific point. This method is especially useful when simpler forms of the functions are needed for computational purposes. Computing the limit often involves dealing with cumbersome expressions, but numerical methods make it easier to handle these by breaking them down into manageable polynomial parts.
For example, approximating \( \log(1 - x) + x e^{x / 2} \) using Taylor series simplifies the limit evaluation process considerably, and this is where numerical analysis shines: reducing complex operations to simpler, polynomial-based calculations.

Polynomial approximations are used to estimate the behavior of functions using polynomials, which are simpler to compute and understand. Taylor polynomials offer such approximations by expressing a function as a sum of polynomials based on its derivatives.
The Taylor polynomial for a function \( f(x) \) around \( a = 0 \) is generally written as: \[T_n(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \cdots + \frac{f^n(0)x^n}{n!}\]Using these, we can approximate \( \cos(x) \) up to the second-order term to compute \(\lim_{x \to 0} \frac{1 - \cos (x)}{x^2} \), simplifying the calculation to one involving straightforward algebra.
This approach is also applied when computing \(\lim_{x \to 0} \frac{\log(1 + x^2)}{2x} \) and \(\lim_{x \to 0} \frac{\log (1-x) + x e^{x / 2}}{x^3}\). By using polynomial approximations, difficult limits become more accessible and easier to evaluate systematically.