Problem 33 - Thomas Calculus

Question

Use series to evaluate the limits. \begin{equation} \lim _{y \rightarrow 0} \frac{y-\tan ^{-1} y}{y^{3}} \end{equation}

Step-by-Step Solution

Answer

The limit is \(\frac{1}{3}\).

Step 1: Expand \\(y- an^{-1} y\\) Using Series

The first step is to recall the series expansion for the function \( an^{-1} y\) around \(y = 0\). The expansion is given by \( an^{-1} y = y - \frac{y^3}{3} + \frac{y^5}{5} - \cdots\). So we subtract this expansion from \(y\) to get:\[y - \tan^{-1} y = y - \left(y - \frac{y^3}{3} + \frac{y^5}{5} - \cdots\right) = \frac{y^3}{3} - \frac{y^5}{5} + \cdots\]

Step 2: Simplify the Limit Expression

Substitute the series expansion from Step 1 into the limit expression:\[\lim _{y \rightarrow 0} \frac{y - \tan^{-1} y}{y^3} = \lim _{y \rightarrow 0} \frac{\frac{y^3}{3} - \frac{y^5}{5} + \cdots}{y^3}\]This simplifies to:\[\lim _{y \rightarrow 0} \left(\frac{1}{3} - \frac{y^2}{5} + \cdots\right)\]

Step 3: Evaluate the Limit as \\(y ightarrow 0\\)

As \(y\) approaches 0, the higher order terms (such as \(-\frac{y^2}{5}\) and beyond) approach 0, leaving us with:\[\frac{1}{3}\]Thus, the limit evaluates to \(\frac{1}{3}\).

Key Concepts

Series Expansion

Series expansion is a mathematical technique used to represent functions as infinite sums of simpler terms. It's particularly useful around points where the function is defined, often called the "expansion point." For functions that involve complex expressions, series expansion helps in simplifying calculations by approximating complex functions with polynomial terms.
In the context of inverse trigonometric functions like \(\tan^{-1} y\), series expansions allow us to have an alternative representation of the function that is easier to handle arithmetically. When \(y\) is near zero, the function \(\tan^{-1} y\) can be expanded into a series like this:

• \(\tan^{-1} y = y - \frac{y^3}{3} + \frac{y^5}{5} - \cdots\)

This is an alternating series, meaning the terms alternatively add and subtract, attribute which helps to converge the series rapidly especially for small values of \(y\).
For small values of \(y\), only the first few terms significantly contribute to the value of the function, which simplifies calculations in evaluating limits.

Taylor Series

The Taylor series is a specific type of series expansion that expresses a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Taylor series are invaluable because they allow complex functions to be expressed in the simplest form possible for a specified neighborhood around a point.
Given a function \(f(y)\), its Taylor series around zero (often referred to as the Maclaurin series in this case) is given by:

• \(f(y) = f(0) + f'(0)y + \frac{f''(0)y^2}{2!} + \frac{f'''(0)y^3}{3!} + \cdots\)

For the inverse tangent function \(\tan^{-1} y\), the Taylor series at \(y = 0\) becomes:

• \(\tan^{-1} y = y - \frac{y^3}{3} + \frac{y^5}{5} - \cdots\)

This series simplifies the calculations necessary for evaluating limits, as substituting higher order terms allows us to approximate behavior of the function around the point of interest. Applying Taylor series often leads to noticeable simplification especially when evaluating complicated limits or integrals.

Inverse Trigonometric Functions

Inverse trigonometric functions are those functions that reverse the action of the standard trigonometric functions like sine, cosine, and tangent. For example, the inverse tangent function, denoted \(\tan^{-1} y\), provides the angle whose tangent is \(y\). These functions are particularly significant in finding angles based on trigonometric ratios.
Inverse functions are useful in various mathematical fields such as calculus, allowing for the integration and differentiation of trigonometric functions. Specifically:

• \(\tan^{-1} y = y - \frac{y^3}{3} + \frac{y^5}{5} - \cdots\)

Through this series representation, we see the effectiveness of connecting inverse trigonometric functions with series expansions. It provides an approximate method for evaluating values that would otherwise be computationally intensive or impractical for direct calculations.
In calculus problems, such as limit evaluation, these expanded forms make it easier to simplify expressions and evaluate terms that contribute insignificantly at approaching limit points.