Problem 29 - Thomas Calculus

Question

L'Hopital's Rule does not help with the limits in Exercises \(27-30 .\) Try it; you just keep on cycling. Find the limits some other way. $$ \lim _{x \rightarrow(\pi / 2)^{-}} \frac{\sec x}{\tan x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify Function Behavior

We need to evaluate the limit \( \lim_{x \rightarrow (\pi / 2)^{-}} \frac{\sec x}{\tan x} \). As \( x \) approaches \( \pi/2 \) from the left, we recognize that \( \sec x = \frac{1}{\cos x} \) becomes very large because \( \cos x \to 0 \) whereas \( \tan x = \frac{\sin x}{\cos x} \) also becomes very large since \( \sin x \to 1 \) and \( \cos x \to 0 \). Thus, both numerator and denominator approach infinity in different increments as \( x \) reaches \( \pi/2 \) from the left.

Step 2: Simplify the Expression

Since both \( \sec x \) and \( \tan x \) involve \( \cos x \), rewrite the expression using trigonometric identities: \[ \frac{\sec x}{\tan x} = \frac{1/\cos x}{\sin x/\cos x} = \frac{1}{\sin x} \]. This reduces the problem to understanding the behavior of \( \frac{1}{\sin x} \) as \( x \to (\pi/2)^{-} \).

Step 3: Evaluate the Limit

Now observe that as \( x \to (\pi/2)^{-} \), \( \sin x \to 1 \). Therefore, \( \frac{1}{\sin x} \to 1 \) as well because the denominator approaches a finite, non-zero constant value. Thus, the limit simplifies to \( 1 \).

Key Concepts

Trigonometric Limits

Trigonometric limits are a fundamental concept in calculus that focus on finding the behavior of trigonometric functions as they approach a certain point. This is essential for understanding how functions behave at specific values. Here, we consider the limit of the function \( \frac{\sec x}{\tan x} \), as \( x \) approaches \( \frac{\pi}{2}^- \), meaning from values less than \( \frac{\pi}{2} \).

- **Secant Function (\( \sec x \))**: Given by \( \sec x = \frac{1}{\cos x} \), this function tends towards infinity as \( x \) approaches \( \frac{\pi}{2} \) due to \( \cos x \to 0 \). - **Tangent Function (\( \tan x \))**: Defined as \( \tan x = \frac{\sin x}{\cos x} \), it also tends towards infinity as \( x \) approaches \( \frac{\pi}{2} \), again because \( \cos x \to 0 \).

These tendencies make direct evaluation complex, prompting a transformation of the expression for easier calculation. Recognizing how these functions behave assists in correctly evaluating trigonometric limits.

Limit Evaluation

Evaluating limits, especially when they involve fractions that approach indeterminate forms like \( \frac{\infty}{\infty} \), can be challenging. In the given problem, using L'Hôpital's Rule might seem tempting but only leads to a recurring cycle without resolution. Instead, simplifying the given trigonometric expression is more effective.

- The expression \( \frac{\sec x}{\tan x} \), when rewritten using trigonometric identities, becomes \( \frac{1}{\sin x} \). This simplifies the problem significantly, as the limit now depends on how \( \sin x \) behaves as \( x \to (\pi/2)^{-} \). - As \( x \) approaches \( \pi/2 \) from the left, \( \sin x \) tends to \( 1 \). Consequently, \( \frac{1}{\sin x} \) also tends toward \( 1 \).
These steps lead us to conclusively determine the limit: \( 1 \). This illustrates the importance of simplification and strategic evaluation in handling limits.

Trigonometry Identities

Trigonometry identities are invaluable tools for transforming complex expressions into simpler forms, making calculus operations like limits easier to evaluate. In this particular problem, identifying suitable trigonometric identities helped bypass direct application of rules like L'Hôpital's, simplifying the solution process.

Key trigonometric identities utilized include:

• \( \sec x = \frac{1}{\cos x} \)
• \( \tan x = \frac{\sin x}{\cos x} \)

By replacing \( \sec x \) and \( \tan x \) in \( \frac{\sec x}{\tan x} \) with these identities, we arrive at \( \frac{1}{\sin x} \). This transformation simplifies the evaluation of trigonometric limits significantly.

Understanding and applying these identities not only aids in solving limits but also enhances comprehension of how various trigonometric functions relate to one another. It underscores the need to look for connections and simpler forms when dealing with complex functions.