Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow 0} \frac{\cos x-1+x^{2} / 2}{x^{4}} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Numerator

Start by simplifying the numerator \( \cos x - 1 + \frac{x^2}{2} \). Use the Taylor expansion for \( \cos x \) around 0: \[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + \cdots \] Substitute into the numerator:\[(1 - \frac{x^2}{2} + \frac{x^4}{24}) - 1 + \frac{x^2}{2} = \frac{x^4}{24}\]So, the numerator simplifies to \( \frac{x^4}{24} \).

Step 2: Substitute into the Original Limit Expression

Substitute the simplified numerator back into the limit, resulting in:\[\lim_{x \to 0} \frac{\frac{x^4}{24}}{x^4}\]

Step 3: Simplify the Limit Expression

Now, simplify the expression from Step 2:\[\lim_{x \to 0} \frac{x^4}{24x^4} = \lim_{x \to 0} \frac{1}{24}\]

Step 4: Calculate the Limit

Since the expression \( \frac{1}{24} \) is constant and does not depend on \( x \), the limit is simply:\(\frac{1}{24}\)

Key Concepts

Taylor Series

The Taylor Series is a powerful tool in calculus that provides an approximation of functions using polynomials. It breaks down a function into its fundamental components around a specific point, usually 0. This helps in simplifying complex functions by expressing them as a sum of easily calculated terms. For the cosine function, the Taylor series around 0 is given by:

• \[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \]

In the current limit problem, we use the Taylor series to expand \( \cos x \) up to the \( x^4 \) term. This simplifies the numerator by eliminating terms that become negligible as \( x \) approaches zero. By substituting the series expansion into the numerator, you can identify and cancel out terms, simplifying the computation of the limit.

Trigonometric Functions

Trigonometric functions like \( \cos x \) play a vital role in calculus, offering unique properties that facilitate the evaluation of limits. The cosine function represents the x-coordinate of a point on the unit circle and oscillates between -1 and 1. When evaluating limits involving trigonometric functions, understanding their behavior around specific points is key.
For example, around zero, \( \cos x \) can be approximated accurately using its Taylor series expansion. This reflects how trigonometric functions can be linked to polynomial equations, which are typically simpler to handle. In this limit evaluation problem, replacing \( \cos x \) with its Taylor expansion reduces the task to a polynomial function, making it easier to solve.

L'Hôpital's Rule

L'Hôpital's Rule is a technique used to evaluate limits of indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Although not directly required in this exercise, understanding it can provide a broader view of how limits can be tackled.
L'Hôpital's Rule states that under certain conditions, the limit of a fraction \( \frac{f(x)}{g(x)} \) can be found by evaluating \( \frac{f'(x)}{g'(x)} \). This is particularly useful when the numerator and denominator both approach zero or infinity as \( x \) approaches a specific point.
For this problem, using Taylor series allowed us to bypass the need for L'Hôpital's Rule by directly simplifying the numerator and denominator leading to a straightforward evaluation, but keep this rule in mind for similar problems you may encounter.

Simplification Techniques

Simplification techniques in calculus are invaluable for making complicated expressions more manageable. These include factoring, expanding polynomial expressions, and substituting known values or identities.

• The original exercise involved simplifying the expression \( \cos x - 1 + \frac{x^2}{2} \) by using the Taylor series expansion of \( \cos x \).
• This allowed the cancellation of terms in the numerator, transforming the original problem into a straightforward evaluation of \( \frac{1}{24} \) after division.

Through simplification, what initially seems complicated becomes clear and easier to work with. This enhances understanding and enables efficient problem-solving. Remember, reducing complexity in mathematical problems often reveals simpler forms which are more intuitive to handle.