$\text { Use a graphing utility to graph the function } f ( x ) = \csc \frac { \pi x } { 2 } \text { and determine the }$ following one-sided limit. $$\lim _ { x \rightarrow 2 ^ { - } } f ( x )$$
A) $- \infty$
B) 2
C) $- 2$
D) $\infty$
E) 0

Question

Quizplus · Accepted Answer

The function $f(x) = \csc\left(\frac{\pi x}{2}\right)$ involves the cosecant function, which is the reciprocal of the sine function. As $x$ approaches 2 from the left, the argument of the cosecant function $\frac{\pi x}{2}$ approaches $\pi$, where the sine function equals 0. Since the cosecant function is the reciprocal of the sine function, $f(x)$ approaches $\infty$ as $x$ approaches 2 from the left, because the sine function approaches 0, making its reciprocal approach infinity.

Use a graphing utility to graph the function f(x)=csc⁡πx2 and determine the \text { Use a graphing utility to graph the function } f ( x ) = \csc \frac { \pi x } { 2 } \text { and determine the } Use a graphing utility to graph the function f(x)=csc2πx​ and determine the

$\text { Use a graphing utility to graph the function } f ( x ) = \csc \frac { \pi x } { 2 } \text { and determine the }$