Problem 18 - Calculus - AP Edition

Question

Find the limits. $$\lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)}$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Recognize the form of the expression

The given limit is \( \lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)} \). As \( x \to 0^+ \), both \( \sin x \) and \( \tan x \) approach \( 0 \), and thus \( \ln(\sin x) \) and \( \ln(\tan x) \) approach \(-\infty\). The limit is an indeterminate form \( \frac{-\infty}{-\infty} \), which requires further analysis.

Step 2: Use small angle approximations

Near \( x = 0 \), use the approximations \( \sin x \approx x \) and \( \tan x \approx x \). Therefore, we have \( \ln(\sin x) \approx \ln(x) \) and \( \ln(\tan x) \approx \ln(x) \).

Step 3: Simplify the expression

Substitute the approximations into the limit expression: \( \lim _{x \rightarrow 0^{+}} \frac{\ln (x)}{\ln (x)} \). This simplifies to \( \lim _{x \rightarrow 0^{+}} 1 \).

Step 4: Evaluate the simplified limit

Since \( \lim _{x \rightarrow 0^{+}} 1 = 1 \), the evaluated limit is 1.

Key Concepts

Indeterminate Forms

When evaluating limits in calculus, sometimes we come across something called an "indeterminate form." This is essentially an expression that doesn't initially give a clear answer when you plug the limiting value into it. Indeterminate forms often arise in limits as fractions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms require special techniques to resolve them, such as algebraic manipulation, L'Hôpital's Rule, or other methods.

• In this exercise, the original expression was \( \lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)} \) which turned into \( \frac{-\infty}{-\infty} \) as both \( \sin x \) and \( \tan x \) approach zero.
• To handle this indeterminate form, you need to simplify the expression further using approximation or calculus rules, which transforms the limit into a solvable form.

Small Angle Approximations

In calculus, small angle approximations are very useful when dealing with trigonometric functions for angles close to zero. These approximations make it easier to simplify expressions.

• For small values of \( x \), \( \sin x \) is approximately equal to \( x \) and \( \tan x \) is also approximately equal to \( x \).
• These approximations are particularly useful in this exercise because they simplify the logarithmic functions of these trigonometric expressions. By rewriting \( \sin x \approx x \) and \( \tan x \approx x \), you transform the original limit expression from a complex form to something that is easier to evaluate.
• Subsequently, the limit expression simplifies from \( \frac{\ln (\sin x)}{\ln (\tan x)} \) to \( \frac{\ln (x)}{\ln (x)} \) which is a straightforward computation.

Natural Logarithm

The natural logarithm, typically written as \( \ln(x) \), is the logarithm to the base \( e \) where \( e \approx 2.718 \, 28 \, 1828 \, 4590 \, 90 \). It is a fundamental concept in calculus and is heavily used in problems involving growth rates, decay, and many physical phenomena.

• In the context of evaluating limits, the properties of the natural logarithm function allow us to manipulate and simplify expressions further.
• Logarithms convert multiplication into addition and division into subtraction, making complex expressions more manageable.
• When \( x \) approaches 0 from the positive side, \( \ln(x) \) tends toward \(-\infty\), a behavior utilized in understanding and simplifying indeterminate forms.
• In our particular problem, recognizing that \( \ln(x) \) tends to \(-\infty\) helped identify the indeterminate form, eventually simplifying the problem to a much easier expression, \( \ln(x)/\ln(x) = 1 \).