Problem 11 - Contemporary Precalculus

Question

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches π from the left is 0.

Step 1: Recognize indeterminate form

As x approaches π, we have \(\cos x \to -1\) and \(\sin x \to 0\). This means that the function becomes $$\frac{0}{1-(-1)} = \frac{0}{2}$$. This is not an indeterminate form, so we can directly find the limit.

Step 2: Directly find the limit

Since we found in the previous step that the form of the function is not indeterminate, we can directly find the limit. As x approaches π from the left, we have $$\lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x} = \frac{0}{2}$$

Step 3: Write the final result

The limit of the given function as x approaches π from the left is $$\lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x} = 0$$

Key Concepts

Indeterminate Form

The concept of an indeterminate form arises in calculus when analyzing the behavior of functions as they approach a specific point. An indeterminate form indicates that we cannot determine the limit of a function as it approaches a certain value just by the function's form at that point.

In our exercise, as the input approaches \( \pi \) from the left, in other words, as \( x \rightarrow \pi^{-} \), the numerator \( \sin x \) approaches 0 while the denominator \( 1 - \cos x \) approaches 2. This leads to a fraction of \( \frac{0}{2} \), which simplifies to 0. This is not an indeterminate form because it results in a finite and distinct limit. Indeterminate forms are more challenging because they require additional techniques to resolve, such as L'Hôpital's rule, algebraic manipulation, or trigonometric identities.

Trigonometric Limits

When evaluating trigonometric limits, certain problems require us to consider the behavior of trigonometric functions as they approach specific angles. Trigonometric limits often need us to utilize trigonometric identities to simplify and evaluate the limit.

However, in the given case of \( \lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x} \), we do not need to employ such techniques, because the function clearly approaches 0 as x gets closer to \( \pi \) from the left. Here, understanding the basic limits of trigonometric functions, such as \( \sin x \) and \( \cos x \) as they approach \( \pi \) gives us the plain answer. Remember, not all trigonometric limits will be this straightforward, and some may involve indeterminate forms which would need further steps to solve.

Calculus

At the heart of calculus is the exploration of functions, limits, derivatives, and integrals. It's a mathematical field that focuses on change and motion, laying the groundwork for a deeper understanding of the natural world and its phenomena.

The limit concept we're examining is a pivotal tool in calculus. It helps mathematicians and scientists understand the behavior of functions at points that are not necessarily within the function's domain or at points where the function exhibits unusual behavior, such as near discontinuities or points of nondifferentiability. Calculus provides the tools, like the one used in this exercise, to meticulously analyze and deduce the behavior of complex functions as they approach any point or infinity.