f(x)=[5x^3-4x^2-8x+9]/[2x^2-1x-3]

Find the equation of the non-vertical asymptote.
y =

Does f(x) intersect its non-vertical asymptote? (yes or no)

What is the smallest value of x at which f(x) intersects its non-vertical asymptote? ( Enter No in the question blank if you answered no above.)

I got the answer to the first two parts, y=2.5x-.75 and yes. But i don't know how to figure out the last part of the question. Please help :)

For Part 3, It would be a question of finding the intersection between the function

f(x) and the oblique asymptote g(x)=5x/2-3/4.
If there is a solution, look for the smallest value of the zeroes.
I found only one at x=27/5.

To find the smallest value of x at which f(x) intersects its non-vertical asymptote, we need to find the x-value at which the function equals its asymptote. In this case, the non-vertical asymptote is given by y = 2.5x - 0.75, which means that f(x) approaches this line as x goes to infinity or negative infinity.

To find the x-value where f(x) intersects its asymptote, we need to set f(x) equal to 2.5x - 0.75 and solve for x. However, notice that f(x) is a rational function, so we should be cautious because it may have some points that are undefined or have a vertical asymptote.

First, let's factor the numerator and denominator to see if there are any common factors we can cancel:

Numerator: 5x^3 - 4x^2 - 8x + 9
Denominator: 2x^2 - x - 3

We can see that there are no common factors to cancel. So, let's proceed to solve for x by setting f(x) equal to its asymptote:

(5x^3 - 4x^2 - 8x + 9) / (2x^2 - x - 3) = 2.5x - 0.75

Notice that both sides of the equation have fractions, so we can multiply through by the common denominator (2x^2 - x - 3) to eliminate the fractions:

(5x^3 - 4x^2 - 8x + 9) = (2.5x - 0.75)(2x^2 - x - 3)

Expand the right side of the equation:

(5x^3 - 4x^2 - 8x + 9) = 5x^3 - 2.5x^2 - 7.5x^2 + 3.75x + 15x - 7.5

Combine like terms:

5x^3 - 4x^2 - 8x + 9 = 5x^3 - 10x^2 + 18.75x - 7.5

Rearrange the equation by moving all terms to one side:

0 = -6x^2 + 26.75x - 16.5

Now, we need to solve this quadratic equation. However, it seems to be a little complicated, so we can use the quadratic formula to find the solutions. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Applying this formula to our equation, where a = -6, b = 26.75, and c = -16.5, we can find the two solutions for x:

x₁ = (-26.75 + √((26.75)^2 - 4(-6)(-16.5))) / (2(-6))
x₂ = (-26.75 - √((26.75)^2 - 4(-6)(-16.5))) / (2(-6))

Evaluating these expressions will give us the two possible x-values at which f(x) intersects its asymptote.