MHB What are the steps to finding an oblique asymptote for a rational function?

  • Thread starter Thread starter mathdad
  • Start date Start date
  • Tags Tags
    Asymptote
Click For Summary
SUMMARY

The oblique asymptote for the rational function f(x) = (x^2 - 16)/(x - 4) is determined to be the line y = x + 4. This conclusion is reached after simplifying the function, which reveals a point discontinuity at (4, 8) due to the denominator being zero at x = 4. The discussion clarifies that long division is necessary when the degree of the numerator is one greater than that of the denominator, leading to the identification of slant asymptotes.

PREREQUISITES
  • Understanding of rational functions
  • Knowledge of asymptotes, specifically oblique and slant asymptotes
  • Familiarity with polynomial long division
  • Concept of point discontinuity in functions
NEXT STEPS
  • Study polynomial long division techniques for rational functions
  • Learn about identifying and graphing point discontinuities
  • Explore examples of rational functions with oblique asymptotes
  • Investigate the differences between horizontal, vertical, and oblique asymptotes
USEFUL FOR

Mathematics students, educators, and anyone studying calculus or analyzing rational functions will benefit from this discussion.

mathdad
Messages
1,280
Reaction score
0
Find the oblique asymptote of f(x) = (x^2 - 16)/(x - 4). I need the steps not the solution.
 
Mathematics news on Phys.org
f(x) simplifies to x + 4, which is the oblique asymptote of f(x).
 
greg1313 said:
f(x) simplifies to x + 4, which is the oblique asymptote of f(x).

I get that the oblique asymptote is a line. Perhaps, I need an example that is a bit more involved.

When is it required for me to use long division to find the oblique asymptote?

Since the oblique asymptote is the line x + 4, what exactly does that mean?
 
The given rational function does not have an oblique asymptote ... it has a point discontinuity at the point (4,8)
 
1. What is a point discontinuity?

2. Where did (4,8) come from?
 
skeeter said:
The given rational function does not have an oblique asymptote ... it has a point discontinuity at the point (4,8)

My mistake. It is equivalent to the line y = x + 4, with a discontinuity at x = 4 (where the denominator is zero).
 
greg1313 said:
My mistake. It is equivalent to the line y = x + 4, with a discontinuity at x = 4 (where the denominator is zero).

You are saying that at x = 4, there is discontinuity. In other words, there is a hole in the graph of the function at the point (4, 8), right?

Why is an oblique asymptote called a slant asymptote?
 
RTCNTC said:
You are saying that at x = 4, there is discontinuity. In other words, there is a hole in the graph of the function at the point (4, 8), right?

Yes

RTCNTC said:
Why is an oblique asymptote called a slant asymptote?

Because the oblique asymptote is a line that is neither horizontal (horizontal asymptote) nor vertical (vertical asymptote), but slanted.
 
This weekend, I will post several rational functions and my solution to each problem.
 

Similar threads

MHB S5.t.5 Find domain and asymptotes.
  • · Replies 3 ·
Replies
3
Views
1K
MHB Horizontal Asymptote of Rational Function
  • · Replies 20 ·
Replies
20
Views
2K
MHB What are the steps for finding asymptotes of rational functions?
  • · Replies 2 ·
Replies
2
Views
2K
High School Understanding exponentiation and logarithms together
  • · Replies 12 ·
Replies
12
Views
2K
MHB What Are the Steps to Find the Horizontal Asymptote of f(x) = (x^2 - 9)/(x - 3)?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Where is the Vertical Asymptote of f(x) = (5 - x^2)/(x - 3)?
  • · Replies 6 ·
Replies
6
Views
2K
MHB *oblique asymptotes of radical expressions
  • · Replies 3 ·
Replies
3
Views
7K
MHB Discontinuity: Asymptotic discontinuity
  • · Replies 4 ·
Replies
4
Views
5K
Oblique Asymptotes: What happens to the Remainder?
  • · Replies 2 ·
Replies
2
Views
17K
MHB Vertical and Horizontal Asymptotes
  • · Replies 3 ·
Replies
3
Views
2K