Problem 6 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{t \rightarrow \pi / 9} \sin (3 t) $$

Step-by-Step Solution

Answer

The limit is \( \frac{\sqrt{3}}{2} \).

Step 1: Understand the Problem

We need to investigate the limit of \( \sin(3t) \) as \( t \) approaches \( \frac{\pi}{9} \). This requires us to observe what happens to the function value near \( \frac{\pi}{9} \).

Step 2: Simplify the Expression

When \( t \) approaches \( \frac{\pi}{9} \), the expression inside the sine function becomes \( 3t = 3 \times \frac{\pi}{9} = \frac{\pi}{3} \). Thus, we are essentially investigating the limit of \( \sin(\frac{\pi}{3}) \).

Step 3: Use Trigonometric Values

The trigonometric value for \( \sin(\frac{\pi}{3}) \) is known: \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \). This provides the result of the limit since this value is constant as \( t \) approaches \( \frac{\pi}{9} \).

Step 4: State the Result Using Limit Notation

Based on the simplification and trigonometric value, the limit can be stated as: \[ \lim _{t \rightarrow \pi / 9} \sin (3 t) = \frac{\sqrt{3}}{2} \].

Key Concepts

Trigonometric Limits

Trigonometric limits often involve understanding the behavior of functions like sine, cosine, and tangent as their variables approach a particular value. Investigating these limits requires evaluating how the trigonometric functions respond as the input values converge to a specific point. In our problem, we are examining the limit of \( \sin(3t) \) as \( t \) moves closer to \( \frac{\pi}{9} \). The trigonometric limit here helps us predict the output of the sine function as the input nears a critical juncture.To handle trigonometric limits:

• Simplify the expression when possible.
• Substitute known trigonometric identities and values.
• Utilize graphs and tables for visual approximation if needed.

In this context, understanding these limits is crucial for solving problems where direct substitution may not be immediately intuitive. With trigonometric identities, one often finds an opportunity to simplify complex expressions, making the limit evaluation process much smoother and more intuitive.

Sine Function

The sine function is one of the most fundamental trigonometric functions. It takes an angle and returns the length of the opposite side of a right triangle divided by the hypotenuse. In mathematical terms, for an angle \( \theta \), this is expressed as \( \sin(\theta) \). The sine function has a periodic behavior, repeating its values in regular intervals of \( 2\pi \).Some essential properties of the sine function include:

• It is continuous and smooth, without breaks or sharp turns.
• The function oscillates between -1 and 1.
• It reaches its maximum at \( \frac{\pi}{2} \) and its minimum at \( \frac{3\pi}{2} \).

In our exercise, by substituting \( 3t = \frac{\pi}{3} \), we see how the sine function's periodicity and known values help us easily compute the limit. Since \( \sin(\frac{\pi}{3}) \) is a standard value equal to \( \frac{\sqrt{3}}{2} \), it makes the investigation straightforward. This knowledge allows us to quickly solve problems using these basic properties of sine.

Limit Investigation

Limit investigation is a process where we explore the behavior of a function as the input approaches a certain value. In calculus, this is often a critical skill necessary for understanding how functions behave, especially in points where they might not be well-defined or intuitive.The steps for investigating a limit typically involve:

• Substitute the input value directly into the function if possible.
• If direct substitution is not possible, simplify the expression.
• Use algebraic manipulation or known values to find the solution.

In our specific example, the limit \( \lim _{t \rightarrow \pi / 9} \sin (3 t) \) is investigated by considering what the expression inside the sine function becomes as \( t \) is near \( \frac{\pi}{9} \). Converting \( 3t = \frac{\pi}{3} \) returns a known value of \( \sin \) that simplifies the process. By understanding both the function's behavior and the trigonometric limits, we reach a precise solution, demonstrating the importance of limit investigation in calculus.