Problem 60 - Precalculus

Question

Evaluate the following limits using a table of values. Given \(g(x)=\left\\{\begin{array}{ll}\sec x & x<\frac{5 \pi}{4} \\ \cot x & x=\frac{5 \pi}{4}, \text { find } \\ \csc x & x>\frac{5 \pi}{4}\end{array}\right.\) a. $$\lim _{x \rightarrow \frac{\pi}{4}} g(x)$$ b. $$\lim _{x \rightarrow \frac{5 \pi}{4}} g(x)$$ c. $$\lim _{x \rightarrow \frac{\pi}{4}} g(x)$$

Step-by-Step Solution

Answer

a. \( \sqrt{2} \); b. \(-\sqrt{2} \); c. \(\sqrt{2}\)

Step 1: Analyze the Function

Given the piecewise function \( g(x) \), divide it into three portions. For \( x < \frac{5\pi}{4} \), \( g(x) = \sec x \). At \( x = \frac{5\pi}{4} \), \( g(x) = \cot x \). For \( x > \frac{5\pi}{4} \), \( g(x) = \csc x \).

Step 2: Calculate Limit as x Approaches \( \frac{\pi}{4} \)

The function is \( g(x) = \sec x \) when \( x < \frac{5\pi}{4} \). Since \( \frac{\pi}{4} < \frac{5\pi}{4} \), we use \( \sec x \) to find \( \lim_{x \to \frac{\pi}{4}} g(x) \). Push values in from both sides and use a calculator to find that \( \sec \frac{\pi}{4} = \sqrt{2} \). So, the limit is \( \sqrt{2} \).

Step 3: Calculate Limit as x Approaches \( \frac{5\pi}{4} \)

We examine the behavior of \( g(x) \) on both sides of \( x = \frac{5\pi}{4} \). As \( x \to \frac{5\pi}{4}^- \), use \( \sec x \). As \( x \to \frac{5\pi}{4}^+ \), use \( \csc x \). Calculate values: \( \sec \frac{5\pi}{4} \approx -\sqrt{2} \) and \( \csc \frac{5\pi}{4} = -\sqrt{2} \). Since both approach the same value, the limit \( \lim_{x \to \frac{5\pi}{4}} g(x) = -\sqrt{2} \).

Step 4: Validate Limit as x Approaches \( \frac{\pi}{4} \) Again

This appears to be a duplicate part of the exercise. We reaffirm using \( Step 2 \) that \( \lim_{x \to \frac{\pi}{4}} g(x) = \sqrt{2} \).

Key Concepts

Limit Evaluation

In mathematics, evaluating limits is fundamental when analyzing how functions behave as they approach specific points. A limit essentially describes what happens to a function as it gets closer to a certain point from either direction. It is expressed as \( \lim_{x \to a} f(x) \) and asks, "What value does \( f(x) \) get close to as \( x \) approaches \( a \)?"When dealing with limits in a piecewise function, you have to consider each segment of the piecewise definition separately. You may find that a function behaves differently depending on whether the point is approached from the left or the right, which is why it's important to look at one-sided limits:

• Left-hand limit: Approached from values less than \( a \) (\( x \to a^- \)).
• Right-hand limit: Approached from values greater than \( a \) (\( x \to a^+ \)).

For overall limit existence, the left and right limits must be equal. If they're equal, the limit exists. If not, the limit does not exist at that point.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are periodic and include functions like sine, cosine, and their reciprocals: secant (\( \sec x \)), cosecant (\( \csc x \)), and cotangent (\( \cot x \)). Here’s a brief overview:

• Secant (\( \sec x \)): The reciprocal of cosine, \( \sec x = \frac{1}{\cos x} \).
• Cosecant (\( \csc x \)): The reciprocal of sine, \( \csc x = \frac{1}{\sin x} \).
• Cotangent (\( \cot x \)): The reciprocal of tangent, \( \cot x = \frac{\cos x}{\sin x} \).

These functions can behave differently depending on the input values, especially when dealing with angles provided in radians. For the given exercise:

• \( \sec \frac{\pi}{4} \) evaluates to \( \sqrt{2} \), since \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \).
• \( \sec \frac{5\pi}{4} \) evaluates to \( -\sqrt{2} \), given cosine symmetry and quadrant considerations.

These transformations require understanding the unit circle and how angles correspond on it.

Calculating Limits Using Tables

One intuitive method to calculate limits is by using tables. When you plug in values of \( x \) that approach a certain point from both sides, a table helps you see the trend or direction in which the function's output values are heading.Here’s how to set up such a table in practice:

• Pick several values of \( x \) close to the point from the left (less than the point) and the right (greater than the point).
• Calculate the function’s value for each \( x \).
• Observe if the values converge to a common number, indicating the limit.

This method complements analytical techniques and provides a clearer picture of how the function behaves near the limit point. In our exercise, you would approach the function for a given segment of \( g(x) \), either \( \sec x \), \( \csc x \), or \( \cot x \), based on the respective intervals noted in the piecewise function, to discern its behavior as \( x \) nears the specific angles \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \).