Question

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$q(s)=\frac{\pi}{s-\sin s}$$

Short answer

Question: Determine the vertical asymptotes of the function $$q(s) = \frac{\pi}{s - \sin s}$$. Answer: To find the vertical asymptotes, we need to determine the values of $$s$$ that make the denominator equal to zero. This involves finding the roots of the equation $$s - \sin s = 0$$. Because of the complexity of this equation, we typically use a graphical approach, such as a tool like Desmos or Geogebra, to observe the points of intersection between the graphs of $$s$$ and $$\sin s$$. Each root found represents a vertical asymptote as a line $$s = a$$. Keep in mind there may be infinitely many intersections, resulting in infinitely many vertical asymptotes.

Step-by-step solution

Identify the denominator of the function

In this case, the denominator is the expression $$s-\sin s$$.

Set the denominator equal to zero

We need to find the values of s that make the denominator equal to zero, so we need to solve the equation: $$s-\sin s=0$$

Analyze the equation

This equation is transcendental, and there is no general algebraic method to find its roots. To find the vertical asymptotes, we can use a graphical approach.

Graph the equation

Using a graphing utility, graph the equation $$s-\sin s=0$$ or $$s=\sin s$$ to find its roots. For example, you may use a tool like Desmos or Geogebra.

Identify the values of s

Observe the points of intersection between the graphs of $$s$$ and $$\sin s$$, which represent the values of s where the function is undefined. These are the roots of the equation $$s=\sin s$$. Keep in mind the graph of $$s=\sin s$$ may have infinitely many intersections, and in that case, there will be infinitely many vertical asymptotes.

Write the vertical asymptotes

For each root found in Step 5, write the vertical asymptote as a line $$s=a$$, where a is the root you found. These lines represent the vertical asymptotes of the function $$q(s)=\frac{\pi}{s-\sin s}$$.