Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a CAS to evaluate the limits. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1-x-x^{2} / 2-x^{3} / 6}{x^{4}} $$

Step-by-Step Solution

Answer

The limit evaluates to \( \frac{1}{24} \).

Step 1: Formulate the Limit Expression

We are tasked with evaluating \( \lim _{x \rightarrow 0} \frac{e^{x}-1-x-x^{2} / 2-x^{3} / 6}{x^{4}} \). This limit appears to employ a Taylor expansion centered around \( x = 0 \) for the function \( e^x \).

Step 2: Recall the Taylor Series Expansion for \( e^x \)

The Taylor series expansion for \( e^x \) around \( x = 0 \) is given by: \[ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots \] This expansion can be used to substitute for \( e^x \) in our limit expression.

Step 3: Substitute and Simplify Expression

Substitute the Taylor expansion into the limit expression:\[ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots \] Now replace \( e^x \) in the limit:\[ \lim _{x \rightarrow 0} \frac{\left(1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots \right) - 1 - x - \frac{x^2}{2} - \frac{x^3}{6}}{x^4} \] The expression simplifies to \( \lim _{x \rightarrow 0} \frac{\left(\frac{x^4}{24} + \cdots\right)}{x^4} \).

Step 4: Evaluate the Simplified Limit

The simplified expression \( \frac{x^4}{24} \) over \( x^4 \) yields \( \frac{1}{24} \). Thus, the limit becomes:\[ \lim _{x \rightarrow 0} \frac{\frac{x^4}{24}}{x^4} = \frac{1}{24} \]

Step 5: Verify with a CAS

Use a computer algebra system (CAS) to evaluate the limit. Input the original limit expression into the CAS and verify that it calculates the result as \( \frac{1}{24} \).

Key Concepts

Taylor Series

The Taylor series is a fundamental concept in calculus that offers a way to represent functions as infinite sums of polynomial terms. Each term represents a derivative of the function evaluated at a single point and is divided by the factorial of the term's degree. This concept helps in approximating complex functions with simpler polynomial expressions.

For example, the Taylor series expansion of the exponential function \( e^x \) about the point 0 is:

• \( e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots \)

Each term is derived from the derivative of \( e^x \) at \( x = 0 \). Notice how the coefficients \( \frac{1}{n!} \) arise from the factorial of the term number, reflecting the accumulated rate of change over each degree.

Using Taylor series, we can approximate \( e^x \) with just a few terms for small values of \( x \). This property was crucial in solving the given limit problem, as we could express \( e^x \) in a form that made the limit evaluation straightforward.

Limits

Limits are a key concept in calculus used to understand the behavior of functions as inputs approach a certain value, often leading to a better understanding of continuous functions and derivatives.

In the problem provided, the use of limits was necessary to evaluate the behavior of the function \( \lim _{x \rightarrow 0} \frac{e^{x}-1-x-x^{2} / 2-x^{3} / 6}{x^{4}} \). The limit signified an infinitesimal change around \( x = 0 \), which allowed us to anticipate the function's value when the input is extremely close to zero.

• To evaluate limits, simplifying expressions is often essential.
• We used the Taylor series to expand \( e^x \) and subtracted lower order terms to highlight higher order behaviors.
• Once simplified, the remaining terms facilitated an easy evaluation as parts of the expression cancelled out, simplifying the limit process and resulting in \( \frac{1}{24} \).

In a sense, limits provide a window into the dynamics of functions at boundary conditions or points of interest, allowing for precise predictions and mathematical analysis.

Exponential Functions

Exponential functions like \( e^x \) are fundamentally important in mathematics due to their unique properties, such as constant growth rates (derivatives) and ubiquitous applications across numerous fields.

The function \( e^x \) is specifically intriguing because:

• Its derivative is itself: \( \frac{d}{dx}e^x = e^x \).
• It features simple compositions in Taylor series for approximations, as shown earlier.

In the problem context, breaking down \( e^x \) into its Taylor series allowed us to comprehend how each small component behaves and therefore facilitated proper computation of limits.

Exponential functions are utilized in real-life phenomena such as compound interest and population growth, due to their nature of growing rapidly over time. The calculus approach enables capturing these scenarios quantitatively and accurately.