Question

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{\phi \rightarrow \pi} \phi \sec \phi $$

Short answer

The limit of the given function as \( \phi \) approaches \( \pi \) is \( -\pi \)

Step-by-step solution

Determine the Approach

The given function is \( \phi \sec \phi \). This is a product of two functions, \( \phi \) and \( \sec \phi \). When calculating the limit, if both these functions have a defined output at the limit value, we can simply substitute the limit value into the function.

Calculate secant at the limit

We first look at \( \sec \phi \), which equals \( \frac{1}{\cos \phi} \). At \( \phi = \pi \), the value of \( \cos \phi \) will be -1. Thus the \( \sec \phi \) will also be -1.

Calculate the Limit

Considering both \( \phi \) and \( \sec \phi \) are defined at \( \phi = \pi \), we substitute \( \pi \) into the function. Therefore, the limit \( \lim_{\phi \rightarrow \pi} \phi \sec \phi = \pi \times -1 = -\pi \)

Key concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics and are widely used in various fields like physics and engineering. Their importance stems from their ability to describe periodic phenomena.
These functions include sine, cosine, tangent, cotangent, secant, and cosecant. Each of these functions has a specific role and relationship to a right triangle or the unit circle. In this context, the cosine function is of interest because it is closely related to the secant function, which we will discuss in the next section.

• The sine function relates the opposite side to the hypotenuse in a right-angle triangle.
• The cosine function deals with the adjacent side to the hypotenuse.
• The tangent function is the ratio of the sine function to the cosine function.

Understanding these basic trigonometric functions helps in comprehending more complex expressions involving secant and other functions.

Secant Function

The secant function, denoted by \( \sec \phi \), is the reciprocal of the cosine function. Therefore, it is defined as \( \sec \phi = \frac{1}{\cos \phi} \).
This means wherever the cosine function gets close to zero, the secant function will approach infinity or become undefined.

• Secant function is crucial when considering the slope of a tangent line in calculus.
• It can represent situations where waves and oscillations are involved, given its periodic nature.
• In the provided exercise, understanding that secant causes a point of interest at \( \phi = \pi \), where \( \cos \phi = -1 \) is important.

Recognizing when the secant function becomes undefined or problematic can help in evaluating limits and determining specific behaviors as variables approach certain values.

Calculating Limits

Limits are a fundamental concept in calculus used to understand the behavior of a function as it approaches a specific value. Calculating limits help us determine the instantaneous rate of change, which is the essence of derivatives.
In the exercise, calculating the limit of \( \phi \sec \phi \) as \( \phi \rightarrow \pi \) involves fixed procedures.

• Firstly, analyze each part of the expression. Here, both \( \phi \) and \( \sec \phi \) are vital.
• Next, find the behavior of the trigonometric function (secant) at the limit value. At \( \phi = \pi \), \( \sec \phi = \frac{1}{\cos \phi} = -1 \).
• Finally, substitute the limit value into the expression: \( \phi \sec \phi \) becomes \( \pi \times -1 = -\pi \).

Thus, the process involves substitution only when the expression is well-defined at the limit value. If it is not clearly defined, more advanced techniques might be necessary.