Answer :

Answer:

To find the asymptotes of the graph

=

2

+

1

y=2x+

x

1

, we need to identify where the function behaves in such a way that it approaches infinity or where it is undefined.

Step-by-step explanation:

Vertical Asymptote:

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not simultaneously equal zero (to avoid holes in the graph).

In

=

2

+

1

y=2x+

x

1

:

The vertical asymptote occurs where

=

0

x=0, because the function is undefined at

=

0

x=0 (division by zero).

Horizontal Asymptote:

Horizontal asymptotes describe the behavior of the function as

x approaches positive or negative infinity.

To find the horizontal asymptote:

Compare the degrees of the numerator and the denominator. Since

2

2x (degree 1) grows faster than

1

x

1

 (degree -1) as

x approaches infinity or negative infinity, the horizontal asymptote is determined by the leading terms.

Therefore, as

x approaches infinity:

=

2

+

1

2

y=2x+

x

1

≈2x

As

x approaches negative infinity:

=

2

+

1

2

y=2x+

x

1

≈2x

Thus, the horizontal asymptote is

=

2

y=2x.

Summary of Asymptotes:

Vertical asymptote:

=

0

x=0

Horizontal asymptote:

=

2

y=2x

These asymptotes describe the behavior of the function

=

2

+

1

y=2x+

x

1

 as

x approaches certain values or infinity.