Problem 34 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{6}\).

Step 1: Expand Inverse Tangent Using Series

Use the series expansion for the inverse tangent function, \[ \tan^{-1} y = y - \frac{y^3}{3} + O(y^5). \] This expansion takes terms up to order \(y^3\) into account.

Step 2: Expand Sine Using Series

Use the series expansion for the sine function, \[ \sin y = y - \frac{y^3}{6} + O(y^5). \] Again, include terms up to order \(y^3\).

Step 3: Calculate the Difference \(\tan^{-1} y - \sin y\)

Subtract the series for \( \sin y \) from the series for \(\tan^{-1} y\): \[ ( y - \frac{y^3}{3} + O(y^5) ) - ( y - \frac{y^3}{6} + O(y^5) ) = - \frac{y^3}{3} + \frac{y^3}{6} + O(y^5). \] Simplify to find: \[ -\frac{y^3}{3} + \frac{y^3}{6} = -\frac{y^3}{6}. \]

Step 4: Expand Cosine Using Series

Use the series expansion for the cosine function: \[ \cos y = 1 - \frac{y^2}{2} + O(y^4). \] Include terms up to order \(y^2\).

Step 5: Evaluate the Limit

Substitute the results from Step 3 into the limit expression:\[ \lim_{y \to 0} \frac{- \frac{y^3}{6}}{y^3 (1 - \frac{y^2}{2} + O(y^4))}. \] Simplify the expression: \[ \frac{- \frac{y^3}{6}}{y^3} = -\frac{1}{6}. \] As \( y \to 0\), higher-order terms become negligible, so the limit evaluates to \(-\frac{1}{6}\).

Key Concepts

Series Expansion

Series expansion is a powerful concept in calculus, particularly useful when evaluating limits. It involves expressing functions as infinite sums of terms calculated from the values of their derivatives at a particular point. This transformation helps simplify complex expressions.

For example, the inverse tangent function, denoted as \( \tan^{-1} y \), can be expanded through a series expansion:

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

This series takes into account terms up to order \(y^3\), allowing us to approximate \( \tan^{-1} y \) accurately near \(y = 0\). Similarly, the sine function, \( \sin y \), can be expanded as:

• \( \sin y = y - \frac{y^3}{6} + O(y^5) \).

Including terms up to \(y^3\) provides a precise approximation for small \(y\). When we subtract these two series, we're left with simpler expressions that are easier to handle in a limit setting.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in calculus. They allow us to determine the angle whose trigonometric function equals a given value. Consider the inverse tangent function, \( \tan^{-1} y \). It represents the angle whose tangent is \(y\).

In our exercise, the inverse tangent function was used in conjunction with series expansion. This combination simplified the calculation of a limit that otherwise would be challenging to solve directly. Importantly, the expansion helps deal with small values of \(y\), which is critical when considering limits approaching zero.

These functions often appear in problems where we need to switch between angles and trigonometric ratios, making them indispensable in solving a range of mathematical problems.

Trigonometric Limits

Trigonometric limits involve evaluating limits that contain trigonometric functions. These limits are crucial in calculus because they appear frequently in various applications, particularly in problems requiring continuity and differentiability.

In the problem at hand, we were dealing with the limit as \( y \to 0 \) of an expression involving \( \tan^{-1} y \), \( \sin y \), and \( \cos y \). This required understanding how each trigonometric function approaches its limit as \( y \) becomes very small.

• The cosine function was expanded as \( \cos y = 1 - \frac{y^2}{2} + O(y^4) \).

Using these expansions, we focused on the leading terms that contribute to the limit, simplifying the expression to evaluate the limit as \( y \to 0 \) efficiently.

When evaluating trigonometric limits, expansions not only simplify the computations but also provide insight into the behavior of functions near certain points, pivotal for deep understanding in calculus.