Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Tell whether each statement is true or false. If false, tell why. The least positive number \(k\) for which \(x=k\) is an asymptote for the cotangent function is \(\frac{\pi}{2}\)

Step-by-Step Solution

Answer

False. The least positive asymptote of \( \cot(x) \) is \( \pi \).

Step 1: Understanding the cotangent function

The cotangent function is defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). It is the reciprocal of the tangent function and is undefined where the sine function is zero.

Step 2: Identify points of discontinuity

The cotangent function is undefined whenever \( \sin(x) = 0 \). The sine of \(x\) equals zero at integer multiples of \( \pi \), therefore these are points of discontinuity for the function.

Step 3: Define asymptotes of the cotangent function

Vertical asymptotes of a function occur where the function goes to infinity, which in the case of \( \cot(x) \), happens at points where \( \sin(x) = 0 \). Since \( \sin(x) = 0 \) at \( x = n\pi \) where \( n \) is an integer, the asymptotes of the \( \cot(x) \) function are located at these points.

Step 4: Determine the first positive asymptote

The first positive value for the asymptote is when \( n = 1 \), i.e., \( x = \pi \). It is the smallest positive value satisfying \( x = n\pi \).

Step 5: Conclusion: Evaluate the statement

The statement is false because the least positive number \( k \) for which \( x = k \) is an asymptote for the cotangent function is \( \pi \), not \( \frac{\pi}{2} \).

Key Concepts

Asymptotes

An asymptote is a line that a curve approaches but never quite touches. It's like a target that the function can get infinitely close to, but can never actually reach. For the cotangent function, vertical asymptotes occur where the function becomes undefined or heads towards infinity. These typically happen when the denominator in a fraction equals zero.

To be specific, for the cotangent function defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), vertical asymptotes occur where \( \sin(x) = 0 \). This makes the function undefined because division by zero is impossible. Consequently, these asymptotes are located at integer multiples of \( \pi \) (e.g., \( x = \pi, 2\pi, 3\pi, \ldots \)).

Understanding asymptotes is crucial because it tells us important information about the behavior of the function over its domain.

Discontinuity

In mathematics, a discontinuity is a point or a set of points where a function is not continuous. For the cotangent function, these points of discontinuity correspond to the locations of the vertical asymptotes.

The cotangent function, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), is discontinuous wherever the sine function is zero. This is because when the sine of \( x \) is zero, the cotangent function cannot be evaluated due to the division by zero issue. Thus, the function has discontinuities at \( x = n\pi \), where \( n \) is an integer.

It's paramount to identify these points so you can accurately sketch the function or predict its behavior in calculus and beyond. Discontinuities tell us the function's break points, which are often crucial in understanding the complete graph.

Trigonometric Functions

Trigonometric functions are fundamental functions in mathematics, used widely in science, engineering, and more to relate angles to ratios of side lengths in right triangles. The primary trigonometric functions include sine, cosine, and tangent. Among others, the cotangent function, defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), is vital as well.

These functions are periodic, meaning they repeat values in regular intervals. This periodicity is useful for modeling waves and oscillations, as seen in physics and engineering applications. They have unique properties, such as specific points of symmetry and various ranges and domains.

• Sine and cosine functions have a period of \( 2\pi \).
• Tangent and cotangent functions have a period of \( \pi \).

Understanding trigonometric functions and their properties can help solve complex geometry problems and model real-world situations effectively.