Question

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{t \rightarrow 0} \frac{\sin 3 t}{2 t} $$

Short answer

The limit of the function \(\lim_{t \rightarrow 0} \frac{\sin 3 t}{2 t}\) is \(\frac{3}{2}\)

Step-by-step solution

Identify the Limit Property

Recognize the limit property \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). This allows us to solve limits of trigonometric functions without resulting in an undefined value.

Manipulate the Function

Try to manipulate the function to match the format of our property. Multiply and divide the function by 3: \n \( \lim_{t \rightarrow 0} \frac{\sin 3 t}{2 t} = \lim_{t \rightarrow 0} \frac{3}{2} *\frac{\sin 3 t}{3 t} \). This way, we can directly apply the limit property.

Apply the Limit Property

Apply the limit property to the manipulated function: \n \( \lim_{t \rightarrow 0} \frac{3}{2} *\frac{\sin 3 t}{3 t} = \frac{3}{2} *1 \). Because \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). Thus, the \(\frac{\sin 3 t}{3 t}\) part of the function equals to 1 as \(t\) approaches 0.

Key concepts

Trigonometric Limits

When students encounter the concept of trigonometric limits, it can often be a source of confusion. A common type of limit they come across involves expressions like \(\lim_{x \rightarrow a} \frac{\sin(bx)}{cx}\). Understanding the sine function's behavior as the variable approaches zero is key to grasping these limits. In our example, we look at \(\lim_{t \rightarrow 0} \frac{\sin 3t}{2t}\), which at first glance may seem complicated due to the constants involved.

However, there's a powerful and standard trick: we aim to express the function in a form that resembles \(\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\), one of the fundamental trigonometric limits. This is why in the solution provided, we manipulate the original expression by multiplying and dividing by the same number to make the expression fit this useful pattern. Therefore, students should always be on the lookout for opportunities to transform complex limits into this much more manageable form.

L'Hopital's Rule

If you've ever been stuck on a limit that results in an indeterminate form like 0/0 or \(\infty/\infty\), L'Hopital's Rule can come to your rescue. This rule states that if the limit of functions \(f(x)\) and \(g(x)\) as \(x\) approaches \(a\) results in an indeterminate form, then \(\lim_{x \rightarrow a} \frac{f(x)}{g(x)}\) can be determined by finding \(\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}\), provided this limit exists.

In the context of our trigonometric limit, L'Hopital's Rule is not necessary, as we don't have an indeterminate form once the function is manipulated. However, for educational purposes, it's good to note that had we encountered a situation where both the numerator and denominator approached zero, L'Hopital's would be a valuable tool for finding the limit.

Sine Function Properties

The sine function, which is sine \(\sin x\), has properties that are extremely useful in calculus, especially when dealing with limits. One fundamental property is that as \(x\) approaches 0, the ratio of \(\sin x\) to \(x\) approaches 1. This might not seem intuitive, but it's a consequence of the sine function's behavior around the origin (0,0) on a graph. It's important for students to remember that this property only holds true as \(x\) approaches 0.

Another relevant property of the sine function that is often leveraged in calculus involves its symmetry and periodicity. The sine function is odd, meaning that \(\sin(-x) = -\sin x\), and it repeats every \(2\pi\) units, known as its period. These insights can be crucial when simplifying expressions involving the sine function in more advanced limit problems.

Limit Properties

One of the cornerstones of understanding limits in calculus is getting to know limit properties. These properties allow us to break down complex limits into simpler components that can be more easily analyzed and resolved. A particularly important property states that the limit of a product is equal to the product of the limits, provided that those limits exist: \(\lim_{x \rightarrow a} [f(x)g(x)] = \lim_{x \rightarrow a} f(x) \times \lim_{x \rightarrow a} g(x)\).

In our textbook example, this property is implicitly used when we separate \(\frac{3}{2}\) from \(\frac{\sin 3t}{3t}\), knowing they can be treated individually. Another property used is that the limit of a constant is just the constant itself. Therefore, with good fundamentals in limit properties, students will find it easier to navigate through seemingly complex limit problems and reach the solutions more efficiently.