Problem 73 - Essential Calculus Early Transcendentals

Question

Find the limit. $$ \lim _{x \rightarrow 0} \ln (\cos x) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Behavior of Cosine Function

First, recognize that as \( x \) approaches 0, \( \cos x \) approaches \( \cos(0) \), which is 1. Therefore, \( \cos x \) is close to 1 and remains positive within a small neighborhood around 0.

Step 2: Use Natural Logarithm Properties

We need the limit of \( \ln(\cos x) \) as \( x \) approaches 0. Since \( \ln(1) = 0 \) and \( \cos x \rightarrow 1 \), if we can express \( \ln(\cos x) \) in a form that can use known limits, it can simplify the process.

Step 3: Expand Cosine Using Taylor Series

\( \cos x \) can be expanded using its Taylor series around 0: \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \ldots \). For \( x \) close to 0, \( \cos x \approx 1 - \frac{x^2}{2} \).

Step 4: Use Logarithmic Approximation

Recall the approximation for natural logarithm \( \ln(1 + u) \approx u \) when \( u \) is close to 0. Here, \( u = -\frac{x^2}{2} \), thus \( \ln(\cos x) \approx \ln(1 - \frac{x^2}{2}) \approx -\frac{x^2}{2} \).

Step 5: Evaluate the Limit

Now calculate\( \lim _{x \rightarrow 0} \left(-\frac{x^2}{2}\right) = -\frac{1}{2} \lim _{x \rightarrow 0} x^2 = 0\) since \( \lim _{x \rightarrow 0} x^2 = 0\).

Step 6: Final Step: Conclude the Limit

The limit is found to be 0, which means \( \lim _{x \rightarrow 0} \ln(\cos x) = 0 \). Therefore, the given limit resolves to 0 as \( x \) approaches 0.

Key Concepts

Taylor Series

In mathematics, the Taylor series allows us to express functions as infinite sums of terms calculated from the values of their derivatives at a single point. This is particularly useful when approximating functions near that point. For example, the Taylor series expansion of the cosine function around 0 is:

• \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots \)

For small values of \( x \), higher order terms become negligible and we can approximate:

• \( \cos x \approx 1 - \frac{x^2}{2} \)

This approximation simplifies calculations in problems involving limits and makes the Taylor series a powerful tool for analysis.
By using these series, complicated functions can be analyzed more easily at "small" values and simplified computations are enabled, as seen in this exercise where we used it to find the limit of \( \ln(\cos x) \).

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.718. The natural logarithm is extensively used in calculus to simplify differentiation and integration processes, especially dealing with exponential functions. A key feature of the natural logarithm that is utilized in limit calculations is:

• \( \ln(1 + u) \approx u \)

This approximation holds true when \( u \) is close to zero. It allows simplification of expressions where \( u \) is a small perturbation from 1. In this specific exercise:

• We used \( u = -\frac{x^2}{2} \)

Converting it to \( \ln(1 + u) \) form helped in evaluating the limit efficiently. This approximation helps streamline calculations and is a fundamental property in learning how limits work with logarithmic functions.

Cosine Function

The cosine function, one of the fundamental trigonometric functions, connects an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. Its analytical nature is especially useful when expressed as a Taylor series. Understanding its behavior as \( x \) approaches a certain value is important. As demonstrated in this exercise, as \( x \) approaches 0:

• \( \cos x \rightarrow 1 \)

This is due to the characteristic limit property of cosine. Analyzing its behavior around small values of \( x \) was crucial for simplifying the expression \( \ln(\cos x) \).
Such trigonometric properties assist in understanding complex expressions and their limitations. Moreover, cosine’s relation within the Taylor series framework broadens our toolbox for handling problems involving small angle approximations and facilitates smooth transitions when working in calculus.