Problem 52 - Essential Calculus Early Transcendentals

Question

Use series to cvaluate the limit. $$\lim _{x \rightarrow 0} \frac{1-\cos x}{1+x-e^{x}}$$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Express Functions as Taylor Series

To evaluate the limit \( \lim _{x \rightarrow 0} \frac{1-\cos x}{1+x-e^{x}} \), start by expressing \(\cos x\) and \(e^x\) as their Taylor series expansions around \( x = 0 \). The Taylor series for \(\cos x\) is:\[\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots\] The Taylor series for \(e^x\) is:\[e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots\]

Step 2: Simplify the Numerator

Using the Taylor series, substitute into \(1-\cos x\):\[1 - \cos x = 1 - \left(1 - \frac{x^2}{2} + \cdots \right) = \frac{x^2}{2} - \cdots\]For small \(x\), this reduces to \( \frac{x^2}{2} \) to leading order.

Step 3: Simplify the Denominator

Now substitute into \(1 + x - e^x\):\[1 + x - \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \right) = -\frac{x^2}{2} - \frac{x^3}{6} - \cdots\]To leading order for small \(x\), this is \(-\frac{x^2}{2}\).

Step 4: Formulate the Limit

Combine the simplified expressions for the numerator and denominator:\[\lim _{x \rightarrow 0} \frac{\frac{x^2}{2}}{-\frac{x^2}{2}}\]This simplifies to:\[\lim _{x \rightarrow 0} \frac{x^2/2}{-x^2/2} = -1.\]

Step 5: Evaluate the Limit

Since both the numerator and denominator simplify to \( \pm \frac{x^2}{2} \), the \(x^2\) terms cancel each other out, leaving the ratio as \(-1\).Thus, the limit evaluates to \(-1\).

Key Concepts

Taylor Series Expansion

The Taylor series expansion is a powerful tool in calculus. It allows us to approximate complex functions with simpler polynomial expressions. These approximations can go as high-degree as necessary for accuracy.
For a function \( f(x) \) around \( x = 0 \), the Taylor series representation is:

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

This expansion assumes the function can be infinitely differentiated. Because terms grow smaller quickly for very small \( x \), only a few leading terms are often needed. This makes Taylor series especially useful for evaluating limits where \( x \) approaches zero.

Cosine Function Series

The Taylor series for the cosine function is a classic example of how a function can be approximated around zero. The series for \( \cos x \) is:

• \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \)

This alternating series starts with 1 and subtracts increasingly smaller powers of \( x \).
Each term is of even degree, reflecting the even symmetry of the cosine function. For small values of \( x \), the higher-order terms become negligible, so \( 1 - \frac{x^2}{2} \) is often sufficient for approximation.
In the limit evaluation we did, this simplification was crucial, as only the leading order term \( -\frac{x^2}{2} \) mattered most.

Exponential Function Series

The exponential function's Taylor series is another commonly used expansion in calculus. For \( e^x \), the series is given by:

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

This series is significant because the powers of \( x \) all have positive coefficients. Its growth is determined by both the power and factorial terms.
When evaluating limits, the higher-order terms become less significant as \( x \) approaches zero. For our limit, simplifying \( e^x \) to \( 1 + x + \frac{x^2}{2} \) was enough. This allowed us to isolate the contributing terms, simplifying the denominator of our limit expression.

Limit Simplification

When dealing with limits, especially with series expansion, simplification is key. By replacing functions with their Taylor series, we focus on the dominant terms.
When \( x \) approaches zero, these dominant or leading order terms determine how the expression behaves. In the original problem, both the numerator and denominator simplified using series:

• Numerator: \( 1 - \cos x \approx \frac{x^2}{2} \)
• Denominator: \( 1 + x - e^x \approx -\frac{x^2}{2} \)

Canceling these gives us the core reason the limit simplifies to \(-1\). The trick is focusing on the significant leading order terms, ensuring unnecessary complexity is avoided.

Leading Order Terms

Leading order terms are the simplest terms that capture the essential behavior of a function around a point. When \( x \rightarrow 0 \), higher powers of \( x \) become almost irrelevant.

• For \( \cos x \), the leading order term is \( -\frac{x^2}{2} \).
• For \( e^x \), terms like \( 1 + x \) are leading compared to \( x^2 \) and higher.

In the analyzed limit, identifying these terms enabled us to reduce complex expressions to simpler, tractable ones.
Leading order terms provide the core insight into how a limit behaves, often making the difference between solving it efficiently or getting stuck in complexity. Understanding them allows students to approach limits with confidence and clarity.