Question 1197629: What are the vertical and a horizontal asymptote at y, if any, for the function f(x)=x^2+x-30 / x^2-2x-15 Found 3 solutions by josgarithmetic, greenestamps, MathLover1:Answer by josgarithmetic(39618) (Show Source):
You can put this solution on YOUR website! For x to extreme left or extreme right, f(x) approaches 1, since the x^2 dominates in both numerator and denominator.
The function also expressed as . Consider what happens near and at . ESPECIALLY at .
To determine the vertical asymptotes, write the function with numerator and denominator in factored form:
The denominator of the fraction can't be 0, so the function value is not defined at x=5 or at x=-3; those two values of x are the potential vertical asymptotes.
Since the factor (x-5) is in both numerator and denominator, the function is equivalent to
everywhere except at x=5. So at x=5 there is a hole in the graph instead of a vertical asymptote.
So there is a vertical asymptote and x=-3, but none at x=5.
The horizontal asymptote of the function is the function value when x becomes very large positive or very large negative. For very large positive or negative values of x, the linear and constant terms become insignificant, so the function value approaches
A graph, showing the given function and its horizontal asymptote...
Note the hole at x=5 won't show up on this graph, because the function is undefined only at that one point. Graphing the given function on a very small interval around x=5 on a good graphing calculator will show the hole in the graph.
Vertical asymptote :
To find the vertical asymptote of a rational function, we simplify it first to lowest terms, set its to, and then solve for values.
the vertical asymptote is
The method to find the horizontal asymptote changes based on the degrees of the polynomials in the numerator and denominator of the function.
If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote!
in your case the polynomials have the degree :
(