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

  • Context: MHB 
  • Thread starter Thread starter mathdad
  • Start date Start date
  • Tags Tags
    Asymptote
Click For Summary

Discussion Overview

The discussion revolves around finding oblique asymptotes for rational functions, specifically focusing on the function f(x) = (x^2 - 16)/(x - 4). Participants seek clarification on the steps involved in identifying oblique asymptotes, the implications of such asymptotes, and related concepts like point discontinuities.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant requests the steps to find the oblique asymptote of the function f(x) = (x^2 - 16)/(x - 4) without seeking the solution itself.
  • Some participants assert that the function simplifies to the line y = x + 4, which they identify as the oblique asymptote.
  • Another participant questions when long division is necessary to find an oblique asymptote.
  • Concerns are raised about the existence of an oblique asymptote, with some participants stating that the function has a point discontinuity at (4, 8).
  • Questions are posed about the nature of point discontinuities and the origin of the point (4, 8).
  • Clarifications are made regarding the equivalence of the function to the line y = x + 4 and the nature of the discontinuity at x = 4.
  • Participants discuss the terminology, questioning why an oblique asymptote is referred to as a slant asymptote.
  • One participant mentions plans to post additional rational functions and their solutions in the future.

Areas of Agreement / Disagreement

There is no consensus on the existence of an oblique asymptote, as some participants argue for its presence while others emphasize the point discontinuity. The discussion remains unresolved regarding the necessity of long division and the implications of the asymptote.

Contextual Notes

Participants express uncertainty about the conditions under which long division is required and the implications of discontinuities in relation to asymptotes. The discussion also reflects varying interpretations of the function's behavior near the discontinuity.

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.
 
Physics 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
Undergrad Asymptotes of Rational Functions....
  • · Replies 3 ·
Replies
3
Views
2K
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
MHB What Are the Steps to Find the Horizontal Asymptote of f(x) = (x^2 - 9)/(x - 3)?
  • · Replies 5 ·
Replies
5
Views
2K
High School Understanding exponentiation and logarithms together
  • · Replies 12 ·
Replies
12
Views
2K
MHB Where is the Vertical Asymptote of f(x) = (5 - x^2)/(x - 3)?
  • · Replies 6 ·
Replies
6
Views
2K
MHB How Do You Find the Range and Oblique Asymptote of \( x + \frac{1}{x} \)?
  • · Replies 6 ·
Replies
6
Views
2K
MHB What is the Slant Asymptote of $\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$?
  • · Replies 3 ·
Replies
3
Views
2K
MHB *oblique asymptotes of radical expressions
  • · Replies 3 ·
Replies
3
Views
7K