Problem 36 - Precalculus with Limits

Question

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to -1}\ \sin\dfrac{\pi x}{2}$$

Step-by-Step Solution

Answer

-1

Step 1: Understanding Limit.

The concept of limit in mathematics references the value that a function or sequence approaches as the input or index approaches some given value. Therefore, when we're asked to find the limit of a function as \(x\) approaches a certain value, then we have to substitute that value into the function and simplify.

Step 2: Apply the Concept of Limit to the Function.

The function where the limit is to be evaluated is given as \(\sin\frac{\pi x}{2}\). Since we want to calculate the limit as \(x\) approaches -1 we replace \(x\) with -1 in the function: \(\sin\frac{\pi(-1)}{2} = \sin(-\frac{\pi}{2})\).

Step 3: Evaluate the Trigonometric Function.

From the unit circle or the sine graph, it's known that \(\sin (-\frac{\pi}{2}) = -1\).

Key Concepts

Trigonometric Limits

When we talk about trigonometric limits, we focus on finding the limit of a function involving trigonometric expressions as the variable approaches a particular value. Here, our task is to determine what happens to the value of the sine function \(\sin \frac{\pi x}{2}\) as \(x\) gets close to \(-1\).
Understanding limits is crucial because they help us determine behaviors of functions at points where they might not be explicitly defined. To find the limit of a trigonometric function:

• Substitute the value into the trigonometric function.
• If substitution leads to a defined value, that's the limit!
• Sometimes, the behavior around the point can help determine the limit even if direct substitution doesn’t work.

This method provides insight into the continuity and behavior of functions at specific points.

Evaluating Trigonometric Functions

Evaluating trigonometric functions can seem tricky, but with practice, it becomes straightforward. In our exercise, after substituting \(x = -1\), we evaluate \(\sin\left(-\frac{\pi}{2}\right)\). To find this value:

• Picture the value on the sine wave or unit circle.
• Recognize that \(\sin\theta\) gives the y-coordinate on the unit circle for angle \(\theta\).
• Since \(\sin\left(-\frac{\pi}{2}\right)\) is the same as looking at a rotation clockwise on the unit circle, it results in \(-1\).

The sine wave crosses the y-axis, creating a simple way to visually grasp trigonometric values quickly. This process not only simplifies evaluation but also helps strengthen the understanding of angles and their trigonometric outputs.

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It's a circle with a radius of 1, centered at the origin of a coordinate system. The angle \(\theta\) in trigonometry corresponds to the distance traveled around the circle. By using the unit circle:

• We determine trigonometric values for any angle.
• The x-coordinate corresponds to \(\cos\theta\), and the y-coordinate corresponds to \(\sin\theta\).
• Remembering specific values like \(\frac{\pi}{2}, \pi, \frac{3\pi}{2}\) can make determining values effortless.

For example, \(\sin\left(-\frac{\pi}{2}\right)\) returns \(-1\), which is the y-coordinate at that angle on the unit circle. Knowing such values enables quick calculations and predictions about function behaviors.