f(x)= (-5x^3 + 3x^2 + 3x -5)/(-8x^2 +2x + 7)
Find the equation of the non-vertical asymptote.
y = ?
Does f(x) intersect its non-vertical asymptote? (yes or no)
If yes, what is the smallest value of x at which f(x) intersects its non-vertical asymptote?
f(x) = 5x/8 - 7/32 + stuff/(-8x^2 +2x + 7)
the line y=5x/8 - 7/32 is the asymptote
graph crosses asymptote at x = -37/10
To find the equation of the non-vertical asymptote, we need to determine the behavior of the function as x approaches positive or negative infinity.
First, let's find the limit of the function as x approaches infinity. To do this, divide each term in the numerator and denominator by the highest power of x, which is x^3 in this case:
f(x) = (-5x^3 + 3x^2 + 3x - 5) / (-8x^2 + 2x + 7)
= (-5 + 3/x - 5/x^3 + 3/x^2) / (-8/x + 2/x^2 + 7/x^3)
As x approaches infinity, all the terms with 1/x or 1/x^2 in the denominator will approach zero. Therefore, the only remaining terms are -5/x^3 and -8/x. As x becomes very large, these terms will also approach zero. This means that the limit of f(x) as x approaches infinity is a finite value.
Next, we find the limit of the function as x approaches negative infinity. Again, divide each term in the numerator and denominator by the highest power of x:
f(x) = (-5x^3 + 3x^2 + 3x - 5) / (-8x^2 + 2x + 7)
= (-5 + 3/x - 5/x^3 + 3/x^2) / (-8/x + 2/x^2 + 7/x^3)
As x approaches negative infinity, all the terms with 1/x or 1/x^2 in the denominator will again approach zero. Therefore, the only remaining terms are -5/x^3 and -8/x. As x becomes very large in the negative direction, these terms will also approach zero. Hence, the limit of f(x) as x approaches negative infinity is also a finite value.
Since the limits of f(x) exist and approach finite values as x approaches positive or negative infinity, the non-vertical asymptote of f(x) exists. The equation of the non-vertical asymptote can be found by taking the limit of f(x) as x approaches infinity or negative infinity.
To check if f(x) intersects its non-vertical asymptote, subtract the equation of the asymptote from f(x) and solve for x to find any possible intersection points. If there are intersection points, identify the smallest value of x at which f(x) intersects its non-vertical asymptote.