Understanding the behavior of functions such as the cosecant (csc) near specific points is fundamental to calculus. As we observe the csc function around the point π, we note a dramatic change. As we approach π from the left, with values like 0.9π, 0.99π, and 0.999π, the csc function decreases, moving towards negative infinity. On the other hand, as we approach π from the right, with values like 1.1π, 1.01π, and 1.001π, the csc function increases, moving towards positive infinity. This difference in approach from either side indicates that the limit of csc(x) as x approaches π is undefined. This concept is a key part of understanding the intriguing properties of trigonometric functions in calculus.
\[ \begin{align*} & \text{Let’s find the limit of } \csc(x) \text{ as } x \text{ approaches } \pi. \\ & \text{Recall that } \csc(x) = \frac{1}{\sin(x)}. \\ & \text{Substitute this in to obtain:} \\ & \lim_{{x \to \pi}} \frac{1}{\sin(x)}. \\ & \text{As } x \text{ approaches } \pi, \sin(x) \text{ approaches } \sin(\pi) \text{ which is 0.} \\ & \text{So, the limit expression becomes:} \\ & \lim_{{x \to \pi}} \frac{1}{0}. \\ & \text{A fraction where the denominator approaches 0 is undefined, hence the limit of } \csc(x) \text{ as } x \text{ approaches } \pi \text{ is undefined.} \end{align*} \]
In the study of calculus, understanding how functions behave as they approach certain points is crucial. One such function is the cosecant function, denoted as csc(x), which is the reciprocal of the sine function. We are particularly interested in the behavior of this function as it approaches π. The following table demonstrates the values of csc(x) at specific points close to π: \[ \begin{array}{c|c} x & \csc(x) \\ \hline 0.9\pi & -1.37 \\ 0.99\pi & -57.30 \\ 0.999\pi & -572.96 \\ 1.001\pi & 572.96 \\ 1.01\pi & 57.30 \\ 1.1\pi & 1.37 \\ \end{array} \] This table shows the behavior of csc(x) as x approaches π from both sides. As we get closer to π from the left (with values like 0.9π, 0.99π, and 0.999π), the value of csc(x) decreases dramatically, heading towards negative infinity. Conversely, as we approach π from the right (with values like 1.1π, 1.01π, and 1.001π), the value of csc(x) increases dramatically, heading towards positive infinity. This tendency towards negative infinity from one direction and positive infinity from the other indicates that the limit of csc(x) as x approaches π is undefined. This is a fundamental concept in calculus and highlights the intriguing properties of trigonometric functions.
The tangent function, denoted as tan(x), is the ratio of the sine to the cosine of the angle x.
As x approaches π/2 from the left, tan(x) increases without bound and goes to positive infinity.
As x approaches π/2 from the right, tan(x) decreases without bound and goes to negative infinity.
This tells us that the limit of tan(x) as x approaches π/2 is undefined, which indicates a vertical asymptote at x = π/2.
Here’s a table that illustrates this:
x
tan(x)
0.4π
1.0
0.49π
2.414
0.499π
57.289
0.501π
−57.289
0.51π
−2.414
0.6π
−1.0
As we can see, as x approaches π/2 from the left, the value of tan(x) gets very large, indicating a trend towards positive infinity. As x approaches π/2 from the right, the value of tan(x) gets very small, indicating a trend towards negative infinity.
To visualize this behavior, we can plot the function. Here’s the updated plot with more points on the graph and the π/2 marking on the x-axis:
