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

[mathdad]

Find the oblique asymptote of f(x) = (x^2 - 16)/(x - 4). I need the steps not the solution.

 

[Greg]

f(x) simplifies to x + 4, which is the oblique asymptote of f(x).

 

[mathdad]

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?

 

[skeeter]

The given rational function does not have an oblique asymptote ... it has a point discontinuity at the point (4,8)

 

[mathdad]

1. What is a point discontinuity?

2. Where did (4,8) come from?

 

[Greg]

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).

 

[mathdad]

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?

 

[Dethrone]

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

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.

 

[mathdad]

This weekend, I will post several rational functions and my solution to each problem.