Problem 28 - Calculus

Question

Use a graph to estimate each of the limits in Exercises \(19-28\) Use radians unless degrees are indicated by \(\theta^{\circ}\). $$\lim _{h \rightarrow 0} \frac{\sin (3 h)}{h}$$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Understanding the Limit

The limit we need to evaluate is \( \lim_{h \rightarrow 0} \frac{\sin(3h)}{h} \). This involves finding how \( \frac{\sin(3h)}{h} \) behaves as \( h \) approaches 0.

Step 2: Use Trigonometric Limit Identity

Recall the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). For our expression involving \( 3h \), we can adapt this identity as \( \lim_{h \to 0} \frac{\sin(3h)}{3h} = 1 \).

Step 3: Manipulate the Expression

Notice that \( \frac{\sin(3h)}{h} = \frac{\sin(3h)}{3h} \times 3 \). Here, the term \( \frac{\sin(3h)}{3h} \) approaches 1 as \( h \rightarrow 0 \) based on the trigonometric limit identity.

Step 4: Apply the Limit

Apply the limit to each part of the expression: \( \lim_{h \rightarrow 0} \frac{\sin(3h)}{3h} \times 3 = 1 \times 3 = 3 \). Thus, the original limit \( \lim_{h \rightarrow 0} \frac{\sin(3h)}{h} = 3 \).

Key Concepts

Limit Evaluation

When evaluating limits, your goal is to determine what happens to a function as its input approaches a certain value. It's like asking, "What does the function get closer to?" In the context of trigonometric limits, we often deal with functions that involve sine or cosine. The specific problem we looked at is \[ \lim_{h \rightarrow 0} \frac{\sin(3h)}{h} \]The main challenge is to understand the behavior of the function as the input \( h \) nears 0. To do this, we make use of known trigonometric limit identities, like \[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]These identities are incredibly useful because they provide us with a quick way to evaluate tricky expressions without needing to graph them every time. By manipulating and adapting the limit identity to fit our problem, we can simplify the expression.

• For instance, by multiplying and dividing by the same expression (e.g., \(3h\)), we maintain the equality while making the limit easier to evaluate.
• This approach ensures that we leverage existing knowledge to simplify complex problems in calculus.

As you work through such exercises, always remember the underlying concepts and identities that can streamline your calculations.

Sine Function

The sine function is a fundamental part of trigonometry, defining the relationship between angles and ratios in a right triangle. In calculus, the sine function comes into play frequently when evaluating limits, especially when the problem involves radii.Sine function properties:

• The sine function is periodic and oscillatory, which means it repeats its values at regular intervals.
• It is also a continuous function, smoothly moving from 0 to 1 and back, which is key for determining limits.

In our problem, we see \[ \sin(3h) \]This indicates that the sine function has been modified by multiplying its argument by 3. As a result, its rate of oscillation changes. But the core principle remains: for very small values of \( h \), the behavior of \( \sin(x) \) approximates that of its argument \( x \). Thus, the manipulation of \( \sin(3h) \) by multiplying \( \frac{1}{3h} \) is a standard approach for aligning the function with known limit identities. Understanding these properties assists in visualizing and simplifying function behavior across different mathematical problems.

Graphical Estimation

Graphical estimation is a fantastic technique for visual learners who want to understand limits without diving directly into formula manipulation. It involves using a graph to visually interpret what happens to a function as the variable approaches a specific value. Let's consider the function in question \[ y = \frac{\sin(3h)}{h} \]Graphing the function around \( h = 0 \) shows us its behavior very clearly. As \( h \) gets closer to 0, you'll notice that the graph of \( y \) approaches a certain height, helping us to infer the limit value.Advantages of graphical estimation:

• It offers an intuition-based approach for those who struggle with algebraic manipulation alone.
• Seeing the actual curve can reinforce your understanding of more abstract calculus concepts.
• This method provides a sanity check against algebraic work, ensuring that your computed limits make sense visually.

Using this method gives you a broader understanding of how functions behave near specific points, integrating intuition with analytical skills in calculus.