91Ó°ÊÓ

Limits are a fundamental concept in calculus and are essential for understanding the behavior of functions as they approach specific points. When dealing with vector-valued functions like \( \mathbf{r}(t) = \langle \cos t, \sin t \rangle \), we evaluate the limit by considering the limits of each component separately. This means we find the limit of \( \cos t \) and \( \sin t \) individually as \( t \) approaches a given value. For instance, when \( t \) approaches \( \frac{\pi}{4} \), we find that both \( \cos \left(\frac{\pi}{4}\right) \) and \( \sin \left(\frac{\pi}{4}\right) \) yield \( \frac{\sqrt{2}}{2} \). Thus, the limit of \( \mathbf{r}(t) \) as \( t \to \frac{\pi}{4} \) is \( \left\langle \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right\rangle \). This approach helps us understand what values the function components are approaching without necessarily meeting them.

Continuity of a function at a point implies that as we approach the point, the value of the function does not exhibit any sudden jumps or breaks. For vector-valued functions like \( \mathbf{r}(t) = \langle \cos t, \sin t \rangle \), a function is considered continuous at a point \( t = a \) if the limit as \( t \to a \) equals the function value at \( t = a \). In the problem, for \( t = \frac{\pi}{3} \), we calculated both the limit and the function value:

• \( \lim_{t \to \frac{\pi}{3}} \mathbf{r}(t) = \left\langle \frac{1}{2}, \frac{\sqrt{3}}{2} \right\rangle \)
• \( \mathbf{r}(\frac{\pi}{3}) = \left\langle \frac{1}{2}, \frac{\sqrt{3}}{2} \right\rangle \)

Since these are equal, we establish that the function \( \mathbf{r}(t) \) is continuous at \( t = \frac{\pi}{3} \). This validates that there are no interruptions or gaps at that point, confirming a smooth transition of function values as \( t \) changes slightly around the point.

Trigonometric functions such as \( \cos t \) and \( \sin t \) play a vital role in defining circular motion and periodic phenomena. For the vector-valued function \( \mathbf{r}(t) = \langle \cos t, \sin t \rangle \), these functions describe every point on the unit circle as \( t \) varies from 0 to \( 2\pi \). Here:

• \( \cos t \) represents the horizontal component (x-axis) and varies from 1 to -1.
• \( \sin t \) represents the vertical component (y-axis) and also ranges from 1 to -1.

Each point \( \langle x, y \rangle \) on the circle corresponds to a specific angle \( t \). At \( t = \frac{\pi}{4} \), both \( \cos \) and \( \sin \) are \( \frac{\sqrt{2}}{2} \), highlighting the 45-degree angle in the first quadrant. Understanding these values helps in graphing trigonometric functions accurately, ensuring correct depiction of the circular trajectory.

Parametric equations allow us to express the coordinates of points that form a curve, using one or more parameters. For example, \( \mathbf{r}(t) = \langle \cos t, \sin t \rangle \) is a parametric equation that maps out a circle. Here, both \( \cos t \) and \( \sin t \) are dependent on \( t \), the parameter representing the angle:

• \( (1, 0) \) corresponds to \( t = 0 \)
• \( (0, 1) \) corresponds to \( t = \frac{\pi}{2} \)
• \( (-1, 0) \) is reached at \( t = \pi \)
• \( (0, -1) \) at \( t = \frac{3\pi}{2} \)

By plotting these points for various \( t \) values, from 0 to \( 2\pi \), you are able to visualize the unit circle. Parametric equations simplify the graphing process by directly indicating how x and y coordinate values change along the curve. This makes them especially useful for representing complex curves, such as ellipses or even more intricate shapes, building a strong foundation for analyzing geometrical figures and motions.