Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \pi / 2}(\tan x-\sec x)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Evaluate the Limit's Form

First, substitute \( x = \frac{\pi}{2} \) into the expression \( \tan x - \sec x \) to determine the form. We have \( \tan\left(\frac{\pi}{2}\right) = \infty \) and \( \sec\left(\frac{\pi}{2}\right) = \infty \). Therefore, \( \tan x - \sec x \) leads to an indeterminate form of \( \infty - \infty \).

Step 2: Rewrite to a 0/0 Form

Rewrite the expression to bring it to a standard indeterminate form suitable for using l'Hôpital's Rule. Notice that \( \tan x - \sec x = \frac{\sin x}{\cos x} - \frac{1}{\cos x} = \frac{\sin x - 1}{\cos x} \). When \( x \rightarrow \frac{\pi}{2} \), both \( \sin x \rightarrow 1 \) and \( \cos x \rightarrow 0 \), leading to a form \( \frac{0}{0} \).

Step 3: Apply l'Hôpital's Rule

Since we have a \( \frac{0}{0} \) form, we can apply l'Hôpital's Rule. Differentiate the numerator and the denominator separately. The derivative of the numerator \( \sin x - 1 \) is \( \cos x \), and the derivative of the denominator \( \cos x \) is \(-\sin x \). Thus, we get:\[\lim _{x \rightarrow \pi / 2} \frac{\cos x}{-\sin x}\]

Step 4: Simplify and Evaluate the Limit

Substitute \( x = \frac{\pi}{2} \) into the new expression:\[\lim _{x \rightarrow \pi / 2} \frac{\cos x}{-\sin x} = \frac{\cos(\frac{\pi}{2})}{-\sin(\frac{\pi}{2})} = \frac{0}{-1} = 0.\] So, the limit evaluates to \( 0 \).

Key Concepts

Indeterminate Forms

When dealing with limits in calculus, it's important to first identify the form of the expression. Indeterminate forms arise when the value of a limit is not immediately clear, often when approaching specific points. These forms do not have a straightforward answer right away and need further manipulation to resolve.
Examples of indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( \infty - \infty \)

In our original exercise, the expression \( \tan x - \sec x \) as \( x \rightarrow \frac{\pi}{2} \) results in \( \infty - \infty \). This is an indeterminate form, meaning that we cannot directly determine the limit's exact value without using methods like rewriting the expression into a "\( \frac{0}{0} \)" form, which is more manageable with calculus techniques like l'Hôpital's Rule.

Limits in Calculus

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach a particular point. Understanding limits helps predict how functions behave as they approach specific values, even if the function isn't explicitly defined at those points.
In the original exercise, we were tasked with finding the limit of \( \tan x - \sec x \) as \( x \) approaches \( \frac{\pi}{2} \). Initially, substituting the value \( x = \frac{\pi}{2} \) leads to an undefined form because both \( \tan \) and \( \sec \) approach infinity. However, using strategies like manipulation and the application of calculus rules helps solve these problems.
Limits thus allow students to explore and solve expressions that aren't immediately apparent by expanding function values to often reveal a hidden pattern. Successfully understanding these concepts is vital for further topics, including derivatives and integrals.

Trigonometric Functions

Trigonometric functions are used often in calculus to describe phenomena involving angles and periodic functions. Functions like sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)) are essential when analyzing circular motion, waves, and other periodic behaviors.
In the context of the original exercise, \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \) were used. Knowing these relationships allows for simplification necessary for calculus solutions. By rewriting the original equation \( \tan x - \sec x \) in terms of \( \sin x \) and \( \cos x \), we enable easier manipulation, ultimately leading,
via l'Hôpital's Rule, to a solvable limit expression. A deep understanding of trigonometric identities helps break complex problems into simpler forms, integral for mastering calculus solutions.

Calculus Problem Solving

Solving calculus problems involves several strategies and mathematics principles. l'Hôpital's Rule is a powerful tool when discussing limits that result in indeterminate forms. Here are the general strategies:

• Identify the limit's form.
• Rewrite expressions as needed to achieve a preferable form for calculus manipulation.
• Apply l'Hôpital's Rule: differentiate the numerator and the denominator separately.
• Simplify and evaluate the expression as \( x \rightarrow a \).

In the given exercise, after establishing the form \( \frac{0}{0} \), differentiation was used to resolve \( \lim _{x \rightarrow \pi/2} \frac{\cos x}{-\sin x} \), simplifying the problem to arrive at a final answer.
Effective calculus problem solving is like solving a puzzle—determine the best approach, apply known techniques or rules, and continually simplify until an answer is clear and within reach.