We have the function:
We must find:
0. the intercepts,
,1. the vertical and horizontal asymptotes.
1) x-intercepts
The x-intercepts are given by the x values such that f(x) = 0. So we must find the values of x that satisfies the equation:
Solving for x, we get:
The real roots of this equation are:
So the x-intercepts are the points:
2) y-intercepts
The y-intercepts are given by the y values such that x = 0. Replacing x = 0 in the definition f(x), we see that the denominator of the second term diverges. So we conclude that there are no y-intercepts.
3) Vertical asymptotes
Vertical asymptotes are vertical lines near which the function grows without bound. From point 2, we know that the function grows without limit when x goes to zero. So one vertical asymptote is:
4) Horizontal asymptotes
Horizontal asymptotes are horizontal lines that the graph of the function approaches when x → ±∞. We consider the limit of the function f(x) when x → ±∞:
We see that the function does not tend to any constant value when x → ±∞. So we conclude that there are no horizontal asymptotes.
5) Oblique asymptotes
When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote.
A function ƒ(x) is asymptotic to the straight line y = mx + n (m ≠ 0) if
We consider the line given by:
We compute the limit:
So we have proven that f(x) has the oblique asymptote:
6) Graph
Plotting the intercepts and the asymptotes, we get the following graph:
Answer
1) x-intercepts: (-0.34975, 0), (0.35757, 0), (15.9922, 0)
2) y-intercepts: none
3) Vertical asymptotes: x = 0
4) Horizontal asymptotes: none
5) Oblique asympsotes: y = x -16
6) Graph
