Questions: Graph the function y=csc x with the window [-6 pi, 6 pi] x[-10,10]. Use the graph to analyze the following limits. a. lim csc x x → 5 x^+ b. lim csc x x → 5 pi^- c. lim csc x x → -5 pi^+ d. lim csc x x → -5 pi^- Graph the function y=csc x. Note that all graphs are shown in a[-6 pi, 6 pi, pi / 2] x[-10,10,1] window. Choose the correct answer below. A. B. C. D.

Graph the function y=csc x with the window [-6 pi, 6 pi] x[-10,10]. Use the graph to analyze the following limits.
a. lim csc x x → 5 x^+
b. lim csc x x → 5 pi^-
c. lim csc x x → -5 pi^+
d. lim csc x x → -5 pi^-

Graph the function y=csc x. Note that all graphs are shown in a[-6 pi, 6 pi, pi / 2] x[-10,10,1] window. Choose the correct answer below.
A.
B.
C.
D.

Transcript text: Graph the function $y=\csc x$ with the window $[-6 \pi, 6 \pi] \times[-10,10]$. Use the graph to analyze the following limits. a. $\lim \csc x$ $x \rightarrow 5 x^{+}$ b. $\lim \csc x$ \[ x \rightarrow 5 \pi^{-} \] c. lim $\csc x$ \[ x \rightarrow-5 \pi^{+} \] d. $\lim _{x \rightarrow-5 \pi^{-}} \csc x$ Graph the function $y=\csc x$. Note that all graphs are shown in $a[-6 \pi, 6 \pi, \pi / 2] \times[-10,10,1]$ window. Choose the correct answer below. A. B. C. D.

Solution

Solution Steps

Step 1: Understanding the Function and Limits

The function given is $ y = \csc x $. The cosecant function, $ \csc x $, is the reciprocal of the sine function, $ \sin x $. Therefore, $ \csc x = \frac{1}{\sin x} $. The function has vertical asymptotes where $ \sin x = 0 $, which occurs at $ x = n\pi $ for any integer $ n $.

Step 2: Analyzing the Limits

We need to analyze the following limits:

$ \lim_{x \to 5\pi^-} \csc x $
$ \lim_{x \to 5\pi^+} \csc x $

Since $ \csc x = \frac{1}{\sin x} $, as $ x $ approaches $ 5\pi $ from the left ($ 5\pi^- $), $ \sin x $ approaches 0 from the positive side, making $ \csc x $ approach $ +\infty $. Conversely, as $ x $ approaches $ 5\pi $ from the right ($ 5\pi^+ $), $ \sin x $ approaches 0 from the negative side, making $ \csc x $ approach $ -\infty $.

Step 3: Graphing the Function

The graph of $ y = \csc x $ will have vertical asymptotes at $ x = n\pi $ and will oscillate between these asymptotes. The function will have a periodic nature with a period of $ 2\pi $.

Final Answer

The correct graph of $ y = \csc x $ in the window $[-6\pi, 6\pi] \times [-10, 10]$ is option A. This graph correctly shows the vertical asymptotes at $ x = n\pi $ and the periodic nature of the cosecant function.

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