f(x)=x^3-3x^2-3x-8/(-3x^2-4x-6)

Find the equation of the non-vertical asymptote.
What is the smallest value of x at which f(x) intersects its non-vertical asymptote?

since for large x, f(x) just looks like x^3/-3x^2, the asymptote is y = -x/3

So, how to find there the curve intersects the asymptote? Duh. Set the two equal:

(x^3-3x^2-3x-8)/(-3x^2-4x-6) = -x/3

It appears they do not intersect. Typo?

To find the equation of the non-vertical asymptote of a rational function, you need to determine the degrees of the numerator and denominator polynomials.

The numerator is x^3 - 3x^2 - 3x - 8, which is a cubic polynomial.

The denominator is -3x^2 - 4x - 6, which is a quadratic polynomial.

Since the degree of the numerator is one more than the degree of the denominator, the non-vertical asymptote will be a slant asymptote.

To find the equation of the slant asymptote, you can perform long division or synthetic division to divide the numerator by the denominator. The quotient obtained will be the equation of the slant asymptote.

Performing long division or synthetic division, we get:

x + 3
---------------
-3x^2 - 4x - 6 | x^3 - 3x^2 - 3x - 8
- (x^3 + 3x^2 + 6x)
--------------
-6x - 8
- (-6x - 9)
-------------
1

So, the equation of the slant asymptote is y = x + 3.

To find the smallest value of x at which f(x) intersects its non-vertical asymptote, we need to find the x-coordinate of the intersection point.

For the non-vertical asymptote, y = x + 3.

Setting f(x) equal to x + 3, we get:

x^3 - 3x^2 - 3x - 8 / -3x^2 - 4x - 6 = x + 3

Multiplying through by -3x^2 - 4x - 6 to eliminate the denominator, we get:

(x^3 - 3x^2 - 3x - 8) = (x + 3)(-3x^2 - 4x - 6)

Expanding and rearranging terms, we have:

x^3 - 3x^2 - 3x - 8 = -3x^3 - 13x^2 - 21x - 18

Combining like terms and simplifying, we get:

2x^3 + 10x^2 + 18x - 10 = 0

To find the smallest value of x at which f(x) intersects its non-vertical asymptote, you can solve this cubic equation for x. However, solving a cubic equation can be complex and involve advanced techniques like factoring or using the cubic formula.

Therefore, you may need to use numerical methods or a graphing calculator to approximate the value of x at which f(x) intersects its non-vertical asymptote.