Consider the functions given by $f ( x ) = 2 \sin x$
and
$g ( x ) = \frac { 1 } { 2 } \csc x$ on the interval (0, $\pi$).
Describe the behavior of each of the functions as x approaches $\pi$ .

A)f approaches 2 and g approaches +$\infty$.
B)f approaches 0 and g approaches +$\infty$.
C)f approaches +$\infty$ and g approaches 0.
D)f approaches 0 and g approaches $\frac { 1 } { 2 }$ .
E)f approaches 2 and g approaches $\frac { 1 } { 2 }$ .

Question

Consider the functions given by   $f ( x ) = 2 \sin x$
and
$g ( x ) = \frac { 1 } { 2 } \csc x$ on the interval (0, $\pi$).
Describe the behavior of each of the functions as x approaches $\pi$  .
 
A)f approaches 2 and g approaches +$\infty$.
B)f approaches 0 and g approaches +$\infty$.
C)f approaches +$\infty$ and g approaches 0.
D)f approaches 0 and g approaches $\frac { 1 } { 2 }$ .
E)f approaches 2 and g approaches $\frac { 1 } { 2 }$ .

Quizplus · Accepted Answer

As $x$ approaches $\pi$, $\sin x$ approaches 0, making $f(x) = 2\sin x$ approach 0. Since $\csc x = \frac{1}{\sin x}$, $g(x) = \frac{1}{2}\csc x$ approaches $+\infty$ because $\sin x$ approaches 0, making the denominator of $\csc x$ approach 0, which causes $g(x)$ to approach $+\infty$.

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Consider the Functions Given By $f ( x ) = 2 \sin x$