Analyze the graph of `f(x)=(x+5)/(x*(x+19))` to determine the equation of the horizontal or oblique asymotote.

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lfryerda's profile pic

lfryerda | High School Teacher | (Level 2) Educator

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If you have a rational function (one polynomial divided by another polynomial), then after you cancel out any common factors between the numerator and denominator you can very quickly determine the asymptotes.  

The vertical asymptotes are always found where the denominator is zero.  In this case, we are looking for the zeros of `x(x+19)` which is at x=0 and x=-19.  The vertical asymptotes are at x=0 and x=-19.

Horizon asymptotes or slant asymptotes are a little trickier.

If the degree of the numerator is less than the degree of the denominator, then there is only a horizontal asymptote at y=0.  That's the case here, since the numerator has degree 1 and the denominator has degree 2.

But, if the degree of the numerator is the same as the degree of the denominator, then there is a horizontal asymptote given by the leading coefficients.  As an example, if you have `g(x)={2x^3-x^2+1}/{5x^3+2}` (degree both 3 here), then there is a horizontal asymptote at `y=2/5`

The 2 is from the numerator and the 5 is from the denominator.  

Finally, we only get slant asymptotes if the degree of the numerator is one larger than the degree of the denominator, like `h(x)={2x^2}/{x+1}` .  Then we use division to find the slant asymptote.

Using division, we see after some work that `2x^2=(x+1)(2x-2)+2` .  And now, we immediately get the slant asymptote `y=2x-2` which came from the quotient in the division statement.

These rules can be quick to remember and will always work.

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The graph of a function f(x)= (x + 5)/(x*(x + 19)) has a horizontal asymptote given by y = 0 as the degree of the numerator is less than that of the denominator.

The slant asymptote is determined by dividing the numerator by the denominator and if the quotient is p(x), the asymptote is y = p(x).

For f(x), there is no slant asymptote as the denominator has the smaller degree than the numerator.

The horizontal asymptote is y = 0, there is no slant asymptote.

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