Problem 31 - Thomas Calculus

Question

Use series to evaluate the limits. \begin{equation} \lim _{t \rightarrow 0} \frac{1-\cos t-\left(t^{2} / 2\right)}{t^{4}} \end{equation}

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{24}\).

Step 1: Recall Maclaurin Series for Cosine Function

The Maclaurin series expansion for the cosine function is given by:\[\cos t = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \frac{t^6}{6!} + \cdots\]This series will be used to approximate the limit.

Step 2: Simplify Expression Using Series

Substitute the Maclaurin series of \(\cos t\) into the given limit expression:\[1 - \cos t - \frac{t^2}{2} = 1 - \left(1 - \frac{t^2}{2} + \frac{t^4}{24} - \cdots \right) - \frac{t^2}{2}\]Simplify to get:\[1 - 1 + \frac{t^2}{2} - \frac{t^4}{24} - \frac{t^2}{2} \]which simplifies to \(-\frac{t^4}{24}\).

Step 3: Evaluate the Limit

Substitute the simplified expression into the limit:\[\lim _{t \rightarrow 0} \frac{-t^4/24}{t^4} = \lim _{t \rightarrow 0} -\frac{1}{24} = -\frac{1}{24}\]The expression simplifies to a constant term as \(t^4\) terms cancel out.

Key Concepts

Series Expansion

Series expansion is a way of representing functions as infinite sums of terms calculated from the values of their derivatives at a single point. This is especially useful for analyzing functions at points where direct evaluation is challenging. The Maclaurin series is a specific type of series expansion, representing a function as a series around the point 0.

The Maclaurin series for a function \( f(t) \) can be expressed as:

• \( f(t) = f(0) + f'(0)t + \frac{f''(0)t^2}{2!} + \frac{f'''(0)t^3}{3!} + \,\cdots\)

In the case of the cosine function, the series focuses on the familiar trigonometric function \( \cos t \), showing how it can be expanded using the series:

• \( \cos t = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \frac{t^6}{6!} + \,\cdots \)

This series is significant for providing precise approximations for small values of \( t \). By employing such series expansions, we can simplify complex expressions or limits related to these functions.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, involving determining the behavior of a function as the variable approaches a specific value. This process is crucial for understanding behavior near points of indeterminacy, such as \lim_{t \rightarrow 0} \, where direct substitution renders an undefined or infinite expression.

To evaluate limits using series expansions, one substitutes the expanded form of a function into the given expression, then simplifies. In the given problem:

• Substitute: The expansion of \( \cos t \) was substituted into the expression \( 1 - \cos t - \frac{t^2}{2} \).
• Simplify: This results in \( -\frac{t^4}{24} \) after cancellation and simplification.

Once the expression is simplified, the limit evaluation becomes straightforward as common terms, like \( t^4 \), cancel.

• The remaining limit, \( \lim_{t \rightarrow 0} \frac{-t^4/24}{t^4} \), simplifies further to the constant \( -\frac{1}{24} \).

Such techniques ensure that we reach an accurate, evaluative conclusion about the function's behavior at specific critical points.

Trigonometric Limits

Trigonometric limits are specific limits involving trigonometric functions such as sine, cosine, and tangent. These often come up in calculus when dealing with derivatives and integrals of trigonometric functions. Understanding how to effectively handle these limits is crucial in solving a wide spectrum of calculus problems.

Typically, these limits require transformations to either simplify them or resolve expressions where direct substitution is ineffective. For instance, using the Maclaurin series simplifies the evaluation of such limits near zero.
In the limit \( \lim_{t \rightarrow 0} \frac{1-\cos t- \frac{t^2}{2}}{t^4} \):

• The series form made it possible to transform the limit expression into a manageable state.
• Handling \( 1 - \cos t \) efficiently is crucial for evaluating trigonometric limits like this.

Understanding trigonometric limits and their evaluations helps in solving other complex functions where \( t \) approaches values that yield indeterminate forms.