The function is $\"\"$ .

Method 1:

To find out the horizontal asymptote of a function use the following hypothesis :

If $\"\"$ then,

1. If $\"\"$ , then the $\"\"$ -axis is the horizontal asymptote.

2. If $\"\"$ , then the horizontal asymptote is the line that is $\"\"$ .

3. If $\"\"$ , then there is no horizontal asymptote. (There is a slant diagonal or oblique asymptote).

Our problem is comes under statement 2, hence

Horizontal asymptote is $\"\"$ , here $\"\"$ and $\"\"$ .

$\"\"$

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

Method 2:

$\"\"$

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

Find the vertical asymptote.

Vertical asymptote appears when the function is not defined.

To find the vertical asymptote, equate denominator of the function to zero.

So $\"\"$ .

$\"\"$ .

Therefore the vertical asymptote is $\"\"$ .

Verification :

Graph :

Graph the function $\"\"$ :

$\"\"$

Observe the graph.

The horizontal asymptote is $\"\"$ and vertical asymptote is $\"\"$ .

The horizontal asymptote is $\"\"$ .

The vertical asymptote is $\"\"$ .