Problem 18 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow-\pi / 3} \frac{3 \theta+\pi}{\sin (\theta+(\pi / 3))}$$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Identify the Indeterminate Form

First, plug in the value of \( \theta = -\frac{\pi}{3} \) into the expression \( \frac{3\theta + \pi}{\sin(\theta + \frac{\pi}{3})} \). Substitute \( \theta = -\frac{\pi}{3} \) which gives us:\[ 3(-\frac{\pi}{3}) + \pi = -\pi + \pi = 0 \]and the denominator:\[ \sin(-\frac{\pi}{3} + \frac{\pi}{3}) = \sin(0) = 0 \]This is an indeterminate form \( \frac{0}{0} \), so we can use L'Hôpital's Rule.

Step 2: Differentiate Numerator and Denominator

Apply L'Hôpital's Rule which involves differentiating the numerator and the denominator separately. Start with the numerator:\( f(\theta) = 3\theta + \pi \)The derivative is:\( f'(\theta) = 3 \)Now, the denominator:\( g(\theta) = \sin(\theta + \frac{\pi}{3}) \)The derivative is:\( g'(\theta) = \cos(\theta + \frac{\pi}{3}) \)

Step 3: Apply L'Hôpital's Rule

With the derivatives, the expression becomes:\[ \lim_{\theta \to -\frac{\pi}{3}} \frac{3\theta + \pi}{\sin(\theta + \frac{\pi}{3})} = \lim_{\theta \to -\frac{\pi}{3}} \frac{3}{\cos(\theta + \frac{\pi}{3})} \]

Step 4: Evaluate the Limit

Now, substitute \( \theta = -\frac{\pi}{3} \) into the new expression:\[ \cos(-\frac{\pi}{3} + \frac{\pi}{3}) = \cos(0) = 1 \]Therefore, the expression simplifies to:\[ \lim_{\theta \to -\frac{\pi}{3}} \frac{3}{1} = 3 \]

Key Concepts

Understanding Indeterminate Forms

In calculus, encountering indeterminate forms like \( \frac{0}{0} \) is common when calculating limits. Indeterminate forms signal that direct substitution in limit calculations won't yield a useful result. Instead, it leads to ambiguous cases that require more sophisticated techniques. With our exercise, when substituting \( \theta = -\frac{\pi}{3} \) into both the numerator and denominator, we obtained zero in each. This is a classic \( \frac{0}{0} \) indeterminate form.

To resolve such forms, mathematicians developed techniques like L'Hôpital's Rule, which allows us to differentiate the numerator and denominator separately. By doing so, we often find that the new expression is much easier to evaluate, as was the case in our problem. After applying L'Hôpital's Rule, we ended up with an expression that no longer bore the \( \frac{0}{0} \) form.

Recognizing indeterminate forms is crucial because they guide us on whether we need to apply special rules, like L'Hôpital's, rather than straightforward substitution.

Grasping Limits

Limits are foundational in calculus, providing a way to understand behavior as values approach a certain point. In our exercise, the goal is to evaluate the behavior of the expression as \( \theta \) approaches \( -\frac{\pi}{3} \).

Limits allow us to explore values that aren't directly accessible or lead to indeterminate forms. By calculating the limit, you're essentially asking, "What value does the expression approach as the input approaches a given point?" This is vital for understanding functions at points where direct evaluation isn't feasible.

In this particular example, L'Hôpital's Rule was used to transform the limit into a form that could be more easily solved. Once we found \( \lim_{\theta \to -\frac{\pi}{3}} \frac{3}{\cos(\theta + \frac{\pi}{3})} \), substitution could be applied straightforwardly because the expression became determinate.

Delving into Derivatives

Derivatives, which measure the rate at which a function changes, play a crucial role in L'Hôpital's Rule. They allow us to understand how the numerator and denominator of a fraction change with respect to each other.

In our exercise, finding the derivative of the numerator \( 3\theta + \pi \) yielded a constant \( 3 \). For the denominator \( \sin(\theta + \frac{\pi}{3}) \), the derivative was \( \cos(\theta + \frac{\pi}{3}) \).

The power of derivatives here is that they transform indeterminate forms into expressions that are easier to evaluate. Once both derivatives are calculated, they replace the original functions in the fraction, giving a new form that can be directly evaluated at the point of interest. This allows us to resolve the indeterminate nature of the original limit problem by analyzing the growth rates of the numerator and the denominator.