Question

Find the asymptote of 3x^2/x(x+3) +1

Answer 1

Vertical asymptote:
`x=-3`

Horizontal asymptote:
`y=4`

Explanation:

Given:

The function is given as:

`f(x)=(3x^2)/(x(x+3))+1`

First, we will simplify the given function.

`f(x)=(3x^2)/(x(x+3))+1\\f(x)=(3x)/(x+3)+(x+3)/(x+3)\\f(x)=(3x+x+3)/(x+3)\\f(x)=(4x+3)/(x+3)`

Now, vertical asymptotes occur when denominator is 0. So,

`x+3=0\\ x=-3`

Therefore, the vertical asymptote is:
`x=-3`

When the degree of numerator is equal to that of the denominator, we have horizontal asymptote equal to
`y=(a)/(b)` where, 'a' is the leading coefficient of numerator and 'b' is the leading coefficient of denominator.

Here, degree of numerator and denominator are same and equal to 1. The value of 'a' is 4 and 'b' is 1. So, horizontal asymptote is given as:

`y=(a)/(b)\\y=(4)/(1)\\y=4`