Chapter 11.4, Problem 7LC

To determine

To explain the difference and similarity between excluded value and a vertical asymptote of a rational function.

Answer to Problem 7LC

Similarity: If an rational function has a vertical asymptote at a point that means that point is also an excluded point of the function.

Difference: An excluded value of the rational function need not be vertical asymptote.

Explanation of Solution

Given:

Excluded value and a vertical asymptote of a rational expression

Concept Used:

1. Vertical asymptote of a rational expression are the vertical line(s) that is(are) correspond to the zeroes of the denominator of rational function.
2. Excluded value(s) of an rational function is(are) the value(s) where the function is not defined.

Calculation:

To explain the similarity between excluded value and a vertical asymptote of a rational function use their corresponding definitions since a vertical asymptote of a rational expression is vertical line that correspond where denominator of rational function is 0, and also the function would not be defined where it’s denominator is 0.

Thus if an rational function has a vertical asymptote at a point that means that point is also an excluded point of the function.

Now let’s explain the difference between excluded value and a vertical asymptote.

Note that it can be possible that function is not defined at a value but it’s denominator need not be 0 at that point, for example the in the rational function

  $\frac{x+3}{\sqrt{x}}$

The excluded values are all x less than 0. Since, $\sqrt{x}$ won’t be defined for negative values of x .

But denominator of this function is not 0 at negative values of x that means x<0 are not vertical asymptote of the given function

Thus, an excluded value of the rational function need not be vertical asymptote.