3.4 Rational function  (Page 3/6)

$$x=c$$

Vertical asymptotes occur at singularity when linear factor in the denominator remains after cancellation or otherwise. Let us investigate three functions given earlier for existence of vertical asymptote.

$$f\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)}$$ $$g\left(x\right)=\frac{{\left(x-1\right)}^2\left(x+2\right)}{\left(x-1\right)\left(x+1\right)}$$ $$h\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{{\left(x-1\right)}^2\left(x+1\right)}$$

For function, f(x), singularities exists at x=1 and -1. Here x=1 is a hole as linear factor (x-1) cancels out completely. The linear factor (x+1), however, does not cancel out. Thus, a vertical asymptote exists at x=-1. See graph shown earlier drawn for f(x).

For function, g(x), also singularities exists at x=1 and -1. Here x=1 is a hole on x-axis as linear factor (x-1) remains in the numerator after cancellation. The linear factor (x+1), however, does not cancel out. Thus, a vertical asymptote exists at x=-1. See graph shown earlier drawn for g(x).

For function, h(x), also singularities exists at x=1 and -1. Here, a vertical asymptote exists at x=1 as linear factor (x-1) remains in the denominator. The linear factor (x+1), however, does not cancel out. Thus, a vertical asymptote also exists at x=-1. See graph shown here for h(x).

Nature of vertical asymptote

The function value assumes large values close to singularity where asymptote exists. The values are directed either in the same of opposite directions. It depends on the polarity of reduced function. If the reduced function has linear factor raised to even power, then values asymptotes in the same direction. On the other hand, if the reduced function has linear factor raised to odd power, then values asymptotes in opposite directions. Let us consider function as defined here,

$$f\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{{\left(x-1\right)}^3\left(x+1\right)}$$

After simplification, the function reduces to :

$$\Rightarrow f\left(x\right)=\frac{\left(x+2\right)}{{\left(x-1\right)}^2\left(x+1\right)}$$

Clearly, (x-1) is raised to even power 2. The graph asymptotes towards large positive values i.e. in the same direction from either side of the asymptote. On the other hand, the linear factor (x+1) is raised to 1 i.e. odd power. Hence function value asymptotes in opposite directions.

Horizontal asymptote

Horizontal asymptote is a horizontal line including x-axis to which graph of function comes closer and closer but never touches. The difference between y-value and asymptote is infinitesimally small for large values of x. An equation of horizontal asymptote has the form,

$$y=c$$

Existence of horizontal asymptote depends on the degree of polynomial in the numerator (n) and degree of polynomial in the denominator (m). There are following three cases :

1: If n>m, then there is no horizontal asymptote. However, if n=m+1, then there exists slant asymptote.

2: If n<m, then x-axis is horizontal asymptote.

3: If n=m, then there is horizontal asymptote exists. In this case, the equation of horizontal asymptote is :

$$y=\frac{\text{Coefficient of highest power term in numeratory}}{\text{Coefficient of highest power term in denominator}}$$