Question

the graph of the function (f(x)=cot x) is given above for the interval (xin0,2pi) only. determine the one - sided limit. then indicate the equation of the vertical asymptote. find (lim_{x
ightarrowpi^{-}}f(x)=) this indicates the equation of a vertical asymptote is (x = ) . find (lim_{x
ightarrow0^{+}}f(x)=) this indicates the equation of a vertical asymptote is (x=) . question help: video message instructor

Explanation:

Step1: Recall cotangent definition

The cotangent function is defined as $\cot x=\frac{\cos x}{\sin x}$.

Step2: Analyze $\lim_{x

ightarrow\pi^{-}}f(x)$
As $x
ightarrow\pi^{-}$, $\cos x
ightarrow - 1$ and $\sin x
ightarrow0^{+}$ (since $\sin x$ is positive for $x\in(0,\pi)$ and approaches 0 as $x
ightarrow\pi$). So, $\lim_{x
ightarrow\pi^{-}}\cot x=-\infty$. The vertical - asymptote occurs when $\sin x = 0$. For $x
ightarrow\pi^{-}$, the equation of the vertical asymptote is $x = \pi$.

Step3: Analyze $\lim_{x

ightarrow0^{+}}f(x)$
As $x
ightarrow0^{+}$, $\cos x
ightarrow1$ and $\sin x
ightarrow0^{+}$ (since $\sin x$ is positive for $x\in(0,\pi)$ and approaches 0 as $x
ightarrow0$). So, $\lim_{x
ightarrow0^{+}}\cot x=\infty$. The vertical - asymptote occurs when $\sin x = 0$. For $x
ightarrow0^{+}$, the equation of the vertical asymptote is $x = 0$.

Answer:

$\lim_{x
ightarrow\pi^{-}}f(x)=-\infty$, $x = \pi$
$\lim_{x
ightarrow0^{+}}f(x)=\infty$, $x = 0$