Problem 64 - Calculus: Early Transcendentals

Question

Use series to evaluate the limit. \( \lim_{x \to 0} \frac {\sqrt {1 + x} - 1 - \frac {1}{2} x }{x^2} \)

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{8}\).

Step 1: Expand the square root using the binomial series

For small values of \(x\), the square root function \(\sqrt{1 + x}\) can be expanded using the binomial series. The expansion is given by \[\sqrt{1 + x} = 1 + \frac{x}{2} - \frac{x^2}{8} + O(x^3)\]. Substitute this into the expression for the limit.

Step 2: Substitute expansion into the original expression

Substitute the expansion of \(\sqrt{1+x}\) into the original limit expression:\[\frac{\sqrt{1+x} - 1 - \frac{1}{2}x}{x^2} = \frac{\left(1 + \frac{x}{2} - \frac{x^2}{8} + O(x^3)\right) - 1 - \frac{1}{2}x}{x^2}.\]Simplify the expression inside the numerator.

Step 3: Simplify the numerator

Simplify the expression inside the numerator:\[\left( 1 + \frac{x}{2} - \frac{x^2}{8} + O(x^3) \right) - 1 - \frac{1}{2}x = -\frac{x^2}{8} + O(x^3).\]The terms \(1\) and \(\frac{x}{2}\) cancel out, leaving only \(-\frac{x^2}{8} + O(x^3)\).

Step 4: Factor and cancel terms

Now that the numerator is \(-\frac{x^2}{8} + O(x^3)\), the expression becomes:\[\frac{-\frac{x^2}{8} + O(x^3)}{x^2}\].Factor out \(x^2\) from the numerator to simplify:\[\frac{-x^2\left(\frac{1}{8} - \frac{O(x^3)}{x^2}\right)}{x^2} = -\left(\frac{1}{8} - \frac{O(x)}{x}\right).\]As \(x\to 0\), \(\frac{O(x)}{x} \to 0\).

Step 5: Evaluate the limit

After simplification, the expression inside the limit becomes \(-\frac{1}{8}\) because the higher order term vanishes as \(x \to 0\). Therefore:\[\lim_{x \to 0} \frac{\sqrt{1+x} - 1 - \frac{1}{2}x}{x^2} = -\frac{1}{8}.\]

Key Concepts

Binomial Series Expansion

The Binomial Series is a powerful tool in calculus, especially when dealing with functions expressed in terms of a square root or other fractional exponents. When we have the function \( \sqrt{1 + x} \), it is not always easy to analyze or take limits directly. This is where the binomial series comes in handy by providing an approximation of the function for small values of \( x \). The relevant expansion for the square root is:

• \( \sqrt{1 + x} = 1 + \frac{x}{2} - \frac{x^2}{8} + O(x^3) \)

This series allows us to express the square root as an infinite series and evaluate limits more effectively as \( x \to 0 \). The higher order terms, denoted as \( O(x^3) \), represent a general expression for the remaining terms that become negligible as \( x \) approaches zero. By substituting this expansion into our expression, we simplify the eventually complex calculations into manageable algebraic terms.

Simplifying Expressions

Simplifying complex expressions into simpler forms is an essential skill in algebra and calculus. When dealing with limits, simplification often involves recognizing terms that cancel out or become insignificant. In the given example, we took:

• The expanded series \( 1 + \frac{x}{2} - \frac{x^2}{8} + O(x^3) \)
• And the terms already present, \( 1 + \frac{1}{2}x \)

By substituting and simplifying, we cancelled out \( 1 \) and \( \frac{x}{2} \). What's left is \( -\frac{x^2}{8} + O(x^3) \), a much simpler form that reveals the main contributing term in the numerator's behavior as \( x \to 0 \). Simplification not only helps in understanding the mathematical process but also in reaching the final evaluative step more directly.

Higher Order Terms

Higher order terms, often represented as \( O(x^n) \), are part of the series that offer insight into the precision of its approximation. In the context of series expansions:

• The term \( O(x^3) \) refers to terms of order \( x^3 \) and higher, which become negligible as \( x \to 0 \).
• These terms are useful because they help determine the primary and secondary effects on the function's value during expansion.

As we evaluate limits, these higher order terms are generally ignored since their impact diminishes significantly due to the smaller magnitude of \( x \). Understanding how these terms play a role allows us to ignore them intelligently in favor of terms that have a significant impact on the result.

Limit of Functions

The process of evaluating the limit of a function like \( \lim_{x \to 0} \frac{\sqrt{1 + x} - 1 - \frac{1}{2} x }{x^2} \) focuses on understanding the approach of function values as \( x \) nears a point.

• By simplifying the function using the binomial series, we've reduced it to manageable parts.
• Factoring further simplifies, allowing expression \( \frac{-x^2 (\frac{1}{8} - \frac{O(x)}{x})}{x^2} \) to directly lead to \( -\frac{1}{8} \).

As \( x \to 0 \), terms like \( \frac{O(x)}{x} \to 0 \), reinforcing the significance of primary terms while dismissing the higher order terms’ effects. This careful breakdown and comprehensive analysis simplify the evaluation of limits, helping demystify the process for students mastering calculus.