Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function.] $$ s(t)=\frac{t}{\sin t} $$

Short answer

The vertical asymptotes of the function \(s(t)=\frac{t}{\sin t}\) are at \(t = n\pi, n \in \mathbb{Z}\)

Step-by-step solution

Analyze the function

First, let's identify the denominator of this function. Here, the denominator is \(\sin t\). The function is undefined when the denominator equals zero.

Solve for when the denominator equals zero

Set \(\sin t=0\) and solve for \(t\). Remember, \(\sin t = 0\) for \(t = n \pi, n \in \mathbb{Z}\), including 0, ±\(\pi\), ±2\(\pi\), etc.

Evaluate the numerator

We should check the numerator at these points. In this case, the numerator is \(t\), hence at \(t = n \pi, n \in \mathbb{Z}\) the function is \(s(t)=\frac{n\pi}{\sin (n\pi)} = \frac{n\pi}{0}\) which is undefined as we expected.

Identify the vertical asymptotes

Therefore, the function \(s(t)=\frac{t}{\sin t}\) has vertical asymptotes at all of the points where \(t = n\pi, n \in \mathbb{Z}\), that is, at all integer multiples of pi.

Key concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are foundational in mathematics, especially when studying periodic phenomena like waves.

In the context of our exercise, we focus on the sine function, denoted as \( \sin t \). The sine function oscillates between -1 and 1, and it is defined for all real numbers \( t \). Understanding the behavior of the sine function is essential when analyzing functions that include it in their formulas, such as \( s(t)=\frac{t}{\sin t} \).

Periodicity and Zeros

The sine function has a period of \( 2\pi \), meaning that its values repeat every \( 2\pi \) radians. Important points to consider are where this function equals zero, specifically at \( t = n \pi, n \in \mathbb{Z} \), which include \( 0, \pm\pi, \pm2\pi \), and so forth. These points, where the sine function equals zero, are crucial in determining undefined points and potential vertical asymptotes in functions involving \( \sin t \).

Limits in Calculus

The concept of limits is central to calculus. It deals with the behavior of a function as the input approaches a certain value. For instance, when we discuss vertical asymptotes, we are essentially talking about the limit of a function as it approaches a certain input value, without ever reaching that value.

Limit notation is expressed as \( \lim_{{t \to c}} f(t) \), meaning we are interested in the behavior of \( f(t) \), as \( t \), gets infinitely close to some constant \( c \).

Finding Vertical Asymptotes

When a function \( f(t) \) increases or decreases without bound as \( t \) approaches some value, the function is said to have a vertical asymptote at that value. In terms of limits, this is represented by \( \lim_{{t \to c}} f(t) = \pm\infty \). To find vertical asymptotes, we often set the denominator of a fraction to zero and solve for \( t \), which indicates the input values for which the function is undefined and might have an asymptote, assuming the numerator does not also equal zero at those points.

Undefined Functions

Undefined functions in mathematics are those that don't have a value at certain points or intervals. A common example occurs when a function has a zero denominator at a specific input value, as division by zero is undefined.

Implications of Undefined Points

In the case of our function \( s(t)=\frac{t}{\sin t} \) from the exercise, we see that wherever the denominator \( \sin t = 0 \) (at multiples of \( \pi \)), the function becomes undefined. This undefined nature creates disruptions in the function's domain and often leads to vertical asymptotes, which graphically represent these undefined points as lines that the function approaches but never crosses or touches.

It is critical, especially for functions involving trigonometric expressions, to analyze the points where the function may be undefined and understand their implications on the graph and behavior of the function.