List all asymptotes (both horizontal and vertical) of the rational function:

f(x)=(10x−3)/(8−7x)

vertical asymptotes when 8-7x=0 solve for x

horizontal asymptote when f(x) does not change, look what happens when x gets very large negative
f(x) appx= 10/7
when very large positive
f(x)=-10/7

To find the asymptotes of a rational function, you first need to determine if there are any vertical asymptotes. Vertical asymptotes occur when the denominator of the rational function is equal to zero. In this case, we need to find the values of x that make the denominator 8 - 7x equal to zero.

To solve the equation 8 - 7x = 0, we can subtract 8 from both sides to isolate the term with x:

-7x = -8

Next, divide both sides by -7 to solve for x:

x = -8 / -7

Simplifying further, -8 / -7 is equal to 8/7. So we have found that there is a vertical asymptote at x = 8/7.

To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator of the rational function. In this case, the numerator is a linear function (degree 1) and the denominator is also a linear function (degree 1).

When the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient is the coefficient of the highest power of x.

In this case, the leading coefficient of the numerator is 10 and the leading coefficient of the denominator is -7.

Therefore, the horizontal asymptote is given by the ratio of the leading coefficients, which is:

y = 10 / -7

Simplifying further, we have y = -10/7. So the horizontal asymptote of the rational function f(x) = (10x - 3) / (8 - 7x) is y = -10/7 and the vertical asymptote is x = 8/7.