Problem 44 - Essential Calculus Early Transcendentals

Question

\(43-44\) . What happens if you try to use l'Hospital's Rule to find the limit? Evaluate the limit using another method. $$\lim _{x \rightarrow(\pi / 2)^{-}} \frac{\sec x}{\tan x}$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Type of Limit

The given limit is \( \lim _{x \rightarrow(\pi / 2)^{-}} \frac{\sec x}{\tan x} \). As \( x \) approaches \( \pi/2 \) from the left, both \( \sec x \) and \( \tan x \) approach infinity, which is an indeterminate form \( \frac{\infty}{\infty} \). This form is suitable for using l'Hospital's Rule.

Step 2: Apply l'Hospital's Rule

To use l'Hospital's Rule, differentiate the numerator and the denominator separately. The derivative of \( \sec x \) is \( \sec x \tan x \), and the derivative of \( \tan x \) is \( \sec^2 x \). Substitute these into l'Hospital's Rule: \[ \lim _{x \rightarrow (\pi/2)^{-}} \frac{\sec x \tan x}{\sec^2 x} = \lim _{x \rightarrow (\pi/2)^{-}} \frac{\tan x}{\sec x} \]. After simplifying, the expression becomes \( \lim _{x \rightarrow (\pi/2)^{-}} \sin x \), which can be evaluated.

Step 3: Evaluate the Simplified Limit

To evaluate \( \lim _{x \rightarrow (\pi/2)^{-}} \sin x \), we acknowledge that as \( x \) approaches \( \pi/2 \) from the left, \( \sin x \) approaches 1 because \( \sin(\pi/2) = 1 \). Therefore, the limit is \( 1 \).

Key Concepts

Limit Evaluation

When evaluating a limit, we explore how a function behaves as the input approaches a certain value. In our exercise, we're looking at the limit of \( \frac{\sec x}{\tan x} \) as \( x \) gets very close to \( \pi/2 \) from the left side. This is written mathematically as \( \lim _{x \rightarrow(\pi / 2)^{-}} \).

The first step is always to identify if a direct substitution of the limit point leads to a meaningful result. If not, we might need other techniques like simplification or mathematical rules, such as l'Hospital's Rule, to find the limit value.

• If direct substitution works (no indeterminate form), simply evaluate the expression at the point.
• If not, check for indeterminate forms that suggest us to manipulate the expression further.

In our case, direct substitution doesn’t work because both \( \sec x \) and \( \tan x \) head towards infinity, creating an indeterminate form.

Trigonometric Limits

Trigonometric functions, which are periodic and bounded, often lead to interesting behavior when evaluating limits. In our exercise, we deal with the trigonometric functions \( \sec x \) and \( \tan x \).

It's important to understand how these functions behave near specific angles:

• \( \sec x = \frac{1}{\cos x} \), turns into infinity as \( x \) approaches \( \pi/2 \) because \( \cos(\pi/2) = 0 \).
• \( \tan x = \frac{\sin x}{\cos x} \), also heads to infinity under the same condition.

By simplifying \( \frac{\sec x}{\tan x} \) through derivative techniques, we transformed a challenging problem into a much simpler one involving \( \sin x \), showcasing the beauty of trigonometric manipulation in limit evaluation.

Indeterminate Forms

Indeterminate forms, like \( \frac{\infty}{\infty} \) or \( \frac{0}{0} \), occur when you try to directly substitute a limit but end up with undefined expressions. In such cases, l'Hospital's Rule is a powerful technique that can help to resolve the expression.

To use l'Hospital's Rule, follow these steps:

• Confirm the function is in an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Differentiating the numerator and denominator independently.
• Re-evaluate the limit of the new function.

In our solution, by applying l'Hospital's Rule, the expression \( \frac{\sec x}{\tan x} \) was reduced to \( \sin x \). Evaluating \( \sin x \) as \( x \) approaches \( \pi/2 \), we achieved a determinate form as it simply approaches 1.