Question

Find the horizontal and vertical asymptotes of​ f(x). ​f(x) equals = StartFraction 6 x Over x plus 2 EndFraction 6x x+2 Find the horizontal asymptotes. Select the correct choice below and fill in any answer boxes within your choice. A. The horizontal​ asymptote(s) can be described by the​ line(s) nothing . ​(Type an equation. Use a comma to separate answers as​ needed.) B. There are no horizontal asymptotes. Find the vertical asymptotes. Select the correct choice below and fill in any answer boxes within your choice. A. The vertical​ asymptote(s) can be described by the​ line(s) negative 2 −2 . ​(Type an equation. Use a comma to separate answers as​ needed.) B. There are no vertical asymptotes.

Answer 1

horizontal asymptote at y = 2, vertical asymptote at x = 1

Explanation:

Answer 2

The horizontal asymptote can be described by the line y = 6

The vertical asymptote can be described by the line x = -2

Explanation:

* Lets the meaning of vertical and horizontal asymptotes

- Vertical asymptotes are vertical lines which correspond to the zeroes

of the denominator of a rational function

- A horizontal asymptote is a y-value on a graph which a function

approaches but does not actually reach

- If the degree of the numerator is less than the degree of the

denominator, then there is a horizontal asymptote at y = 0

- If the degree of the numerator is greater than the degree of the

denominator, then there is no horizontal asymptote

- If the degree of the numerator is equal the degree of the denominator,

then there is a horizontal asymptote at y = leading coefficient of the

numerator ÷ leading coefficient of the denominator

* Lets solve the problem

∵
`f(x)=(6x)/(x+2)`

∵ The numerator is 6x

∵ The denominator is x + 2

∴ The numerator and the denominator have same degree

∵ The leading coefficient of the numerator is 6

∵ The leading coefficient of the denominator is 1

∴ There is a horizontal asymptote at y = 6/1

∴ The horizontal asymptote can be described by the line y = 6

- Put the denominator equal zero to find its zeroes

∵ The denominator is x + 2

∴ x + 2 = 0

- Subtract 2 from both sides

∴ x = -2

∴ The vertical asymptote can be described by the line x = -2