Problem 45 - Thomas Calculus

Question

Limits with trigonometric functions Find the limits in Exercises \(43-50 .\) $$\lim _{x \rightarrow 0} \sec x$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Secant Function

The secant function can be expressed in terms of the cosine function as \( \sec x = \frac{1}{\cos x} \). To find \( \lim_{x \to 0} \sec x \), we have to consider the behavior of \( \cos x \) as \( x \) approaches 0.

Step 2: Find \( \lim_{x \to 0} \cos x \)

We know from trigonometric limits that \( \lim_{x \to 0} \cos x = \cos(0) = 1 \). This is because cosine is a continuous function, so the limit of \( \cos x \) as \( x \to 0 \) equals the value of the function at that point.

Step 3: Apply Limit to Secant Function

Given that \( \sec x = \frac{1}{\cos x} \), substitute the limit we found: \( \lim_{x \to 0} \sec x = \lim_{x \to 0} \frac{1}{\cos x} = \frac{1}{\lim_{x \to 0} \cos x} = \frac{1}{1} \).

Step 4: Conclude the Limit

Since \( \frac{1}{1} = 1 \), the limit of the secant function as \( x \) approaches 0 is 1. Therefore, \( \lim_{x \to 0} \sec x = 1 \).

Key Concepts

Secant Function

The secant function is a trigonometric function symbolized by \( \sec x \). It is defined as the reciprocal of the cosine function, which means:

• \( \sec x = \frac{1}{\cos x} \)

This relationship indicates that wherever the cosine is zero, the secant function will be undefined, because division by zero is not possible.
This makes understanding the secant function reliant on grasping the behavior of the cosine function, particularly at key points like \( x = 0 \).
Since the cosine of zero is 1, which is not zero, the secant function is well defined at this point.
Understanding these relationships helps when evaluating limits involving the secant function.
Being able to transition smoothly from one trigonometric function to another by using their reciprocal relationships is crucial in solving trigonometric limits effectively. When dealing with trigonometric limits, the secant function can be particularly tricky because it magnifies any behavior of the cosine function as it approaches its limits. This is something to be especially aware of when the cosine function's value deviates from being precisely 1 or -1. This amplification can cause the secant function to have larger values and greatly impact the limit.

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the foundational trigonometric functions. It describes the x-coordinate of a point on the unit circle as the angle \( x \) varies.
The key feature of the cosine function is its continuous nature, which means it does not have sudden jumps or breaks. Because of this, certain limits involving \( \cos x \) are straightforward, like \( \lim_{x \to 0} \cos x = \cos(0) = 1 \). This continuity makes cosine an ideal candidate for substituting into other functions to evaluate limits through direct substitution.

• The cosine function fulfills the property \( -1 \leq \cos x \leq 1 \) for any real number \( x \).
• At \( x = 0 \), the cosine value is 1, which is often pivotal in evaluating limits involving reciprocal trigonometric functions like secant.

The cosine function's behavior around zero is crucial because it remains well-behaved (non-zero).
This plays an important role in reciprocal functions like \( \sec x = \frac{1}{\cos x} \) by allowing them to remain defined and behave predictably when \( x \) is close to 0. This continuity guarantees the foundational stability of numerous trigonometric equations and identities.

Limits and Continuity

Limits and continuity are foundational concepts in calculus, crucial for understanding trigonometric limits. A limit is the value a function approaches as the input approaches a certain point. Continuity means the function is smooth and unbroken; the value the function approaches as the input nears a point is the actual value at that point.In the limit \( \lim_{x \to 0} \sec x \), we first express the secant function in terms of cosine. Because \( \sec x = \frac{1}{\cos x} \), we need to evaluate the limit of cosine as \( x \) approaches 0. Since the cosine is continuous, \( \lim_{x \to 0} \cos x = \cos(0) = 1 \), thus \( \frac{1}{\cos x} \) smoothly approaches \( \frac{1}{1} = 1 \).
The calculations show that as long as \( \cos x \) does not reach 0, \( \sec x \) can be evaluated using the inverse of the cosine limit.

• Continuity ensures that small changes in \( x \) lead to small changes in \( \cos x \), which translates directly to small changes in \( \sec x \).
• In continuity, the function is essentially predictable at the point, which is critical for evaluating limits by substitution.

Thus, continuity allows us to treat the limit at a specific point as the value of the function at that point, making the limit of the secant at 0 simple to determine. This understanding of how limits work heavily influences problem-solving techniques for trigonometric functions.