Chapter 1.2, Problem 67E

Solution

To find: the functions have graphs that do intersect their horizontal asymptotes.

The function $f\left(x\right)=\frac{x}{x^2-1}$ , $h\left(x\right)=\frac{x^2}{x^3+1}$ and $g\left(x\right)=\frac{x}{x^2-a}$ have graphs that do intersect their horizontal asymptotes.

Given information: 

The Greek roots of the word asymptote mean not meeting, since graphs tend to approach, but not meet, their asymptotes.

The function $f\left(x\right)=\frac{x}{x^2-1}$ , $g\left(x\right)=\frac{x}{x^2-a}$ and $h\left(x\right)=\frac{x^2}{x^3+1}$ .

Calculation:

Consider the function

  $f\left(x\right)=\frac{x}{x^2-1}$

The function has a horizontal asymptote ( $x$ approaches $0$ ) in both the positive and

negative directions; the function also crosses the $x-$ axis at $\left(0,0\right)$ .

  

Yes, functions have graphs that intersect their horizontal asymptotes.

Consider the function

  $g\left(x\right)=\frac{x}{x^2-a}$

The function has a horizontal asymptote ( $x$ approaches 0 $)$ in both the positive and

negative directions; the function also crosses the $x-$ axis at $\left(0,0\right)$ .

  

Yes, the functions have graphs that intersect their horizontal asymptotes.

Consider the function

  $h\left(x\right)=\frac{x^2}{x^3+1}$

Let $y=h\left(x\right)$ .

Putting the value of $x=0$ then $y=0$ .

And $x=1$ , $y=0.5$

And $x=-1$ , $y=\infty$

Plot the graph:

  

The function has a horizontal asymptote at $y=0$ , but intersects that asymptote at $\left(0,0\right)$

Yes, functions have graphs that intersect the horizontal asymptote.