$\"\"$

The function is $\"\"$ , $\"\"$ .

Horizontal asymptotes are horizontal lines that the graph of the function

approaches as $\"\"$ tends to $\"\"$ or $\"\"$ .

The horizontal asymptote is $\"\"$ .

From exercise (51): $\"\"$ .

Therefore, the horizontal asymptote is $\"\"$ .

$\"\"$

Find the relative extrema.

Consider $\"\"$ .

$\"\"$

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

Find the critical points, by equating $\"\"$ .

$\"\"$

$\"\"$ cannot be zero.

$\"\"$

$\"\"$ .

The critical number is $\"\"$ .

$\"\"$

Consider the test intervals to find the interval of increasing and decreasing.

Test intervals are $\"\"$ and $\"\"$ .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

The function $\"\"$ is increasing on the interval $\"\"$ .

The function $\"\"$ is decreasing on the interval $\"\"$ .

$\"\"$

$\"\"$ changes from positive to negative.

Therefore according to First derivative test, the function has maximum at $\"\"$ .

When $\"\"$ , $\"\"$ .

Therefore, the relative maximum is $\"\"$ .

Graph the function $\"\"$ .

$\"\"$

Horizontal asymptote is $\"\"$ .

The relative maximum is $\"\"$ .

Graph of the function $\"\"$ .

$\"\"$