function\ntangent function:\n$f(x)=\\tan x$\nleft end behavior (use limit notation):\nright end behavior…

Explanation:

Step1: Recall properties of tangent function

The tangent function $y = \tan x=\frac{\sin x}{\cos x}$.

Step2: Determine intervals of increasing

The tangent function $y = \tan x$ is increasing on intervals $\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right)$ for $k\in\mathbb{Z}$.

Step3: Determine intervals of decreasing

The tangent function has no intervals of decreasing.

Step4: Find left - end behavior

As $x\to-\infty$, the function $\tan x$ does not have a simple limit. But if we consider the periodic nature, we can say that as $x\to-\frac{\pi}{2}+k\pi^{-}$, $\lim_{x\to-\frac{\pi}{2}+k\pi^{-}}\tan x=-\infty$.

Step5: Find right - end behavior

As $x\to\infty$, as $x\to\frac{\pi}{2}+k\pi^{+}$, $\lim_{x\to\frac{\pi}{2}+k\pi^{+}}\tan x=\infty$.

Step6: Determine horizontal asymptotes

Since the range of $y = \tan x$ is $(-\infty,\infty)$, there are no horizontal asymptotes.

Step7: Determine vertical asymptotes

The function $y = \tan x$ has vertical asymptotes at $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$ because $\cos x = 0$ at these points.

Step8: Find domain

The domain of $y=\tan x$ is all real numbers except $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$, which in interval notation is $\bigcup_{k\in\mathbb{Z}}\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right)$.

Step9: Find range

The range of the tangent function is $(-\infty,\infty)$.

Answer:

• Intervals of increasing: $\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right),k\in\mathbb{Z}$
• Intervals of decreasing: None
• Left - end behavior: As $x\to-\frac{\pi}{2}+k\pi^{-},\lim_{x\to-\frac{\pi}{2}+k\pi^{-}}\tan x = -\infty$
• Right - end behavior: As $x\to\frac{\pi}{2}+k\pi^{+},\lim_{x\to\frac{\pi}{2}+k\pi^{+}}\tan x=\infty$
• Horizontal asymptotes: None
• Vertical asymptotes: $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$
• Domain: $\bigcup_{k\in\mathbb{Z}}\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right)$
• Range: $(-\infty,\infty)$