Oblique asymptote (1 Viewer)

[kevin101]

hey
whats the oblique asymptote for y=x^3/ x+1
The answer is supposed to be y=x^2-x+1

O yea working please. Thanks

 

[xV1P3R]

You would do this by long dividing x³ by (x+1), but since I don't know how to type it up here, i'll do it the long way
y = x³/(x+1)
y = (x³ + x² - x²)/(x+1)
y = x² - (x²)/(x+1)
y = x² - (x² + x - x)/(x+1)
y = x² - x + (x)/(x+1)
y = x² - x + (x+1-1)/(x+1)
y = x² - x + 1 - 1/(x+1)........................You can get this result much quicker with long division
Whatever isn't a part of the fraction is your asymptote. You can do this with any function eg.
y = x/(x+1)
y = (x+1-1)/(x+1)
y = 1 - 1/(x+1)
Giving you a horizontal asymptote of y = 1

 

[Drongoski]

hey
whats the oblique asymptote for y=x^3/ x+1
The answer is supposed to be y=x^2-x+1

O yea working please. Thanks

that's not called an oblique asymptote

if u have, say: y = 2x^3 -3x^2 +5 /(x^2-2x)

= 2x + 1 + (2x+5)/(x^2-2x)

then u have an oblique (slant) asymptote: y = 2x + 1

The asymptote is a straight line

To have an oblique asymptote, you need the numerator polynomial to have a degree of 1 higher than that of the denominator polynomial.

 

[kevin101]

ok i get it. thanks
I usually use limits and then i divide by the highest power. but it didnt work for this question.

 

[cutemouse]

That wouldn't work. You'd need to divide it out first.

 

[jet]

ok i get it. thanks
I usually use limits and then i divide by the highest power. but it didnt work for this question.

That is only useful if the degree of the denominator ≥ degree of the numerator.

 

[kevin101]

ok i see. so if the highest degree of the denominator is greater than or equal to the highest degree of the numerator i should use the limits method

and if the highest degree of the numerator is greater than the highest degree on the denominator i use long division.

 

[gurmies]

You could also do this:



Pardon me jetblack2007 - my LaTeX isn't working at all =/

 

[cutemouse]

There also another way by "inspection" to determine horizontal asymptotes.

If the degree on the numerator is equal to the degree on the denominator then a horizontal asymptote exists.

Its equation is:

 

[jet]

I'm going to move this to the 4-unit forums.

 

[cutemouse]

Why? This stuff is well within the scope of the Ext 1 Maths course...

 

[addikaye03]

Why? This stuff is well within the scope of the Ext 1 Maths course...

The above method displayed by gurmies is 4U technique, not 3U.

 

[cutemouse]

Yes but that was just an aside. The topic is about finding oblique asymptotes in curve sketching, which is a part of the Extension 1 course.

 

[jet]

Actually, the 3 unit syllabus only ever mentions horizontal and vertical asymptotes, never oblique asymptotes, and I have never found a 3 unit question which asks for oblique asymptotes.

 

[cutemouse]

Actually, the 3 unit syllabus only ever mentions horizontal and vertical asymptotes, never oblique asymptotes, and I have never found a 3 unit question which asks for oblique asymptotes.

I would still recommend 3U students to go through this because I've seen a few questions asked in assessments and trials of schools.

However, I haven't ever seen them in 4U exams.

 

[Carrotsticks]

Actually, the 3 unit syllabus only ever mentions horizontal and vertical asymptotes, never oblique asymptotes, and I have never found a 3 unit question which asks for oblique asymptotes.

The Cambridge 3U book Year 11 introduces the concept of oblique asymptotes using limits. Although it is not relevant, it teaches it nonetheless.

 

[cutemouse]

Interesting. Specifically how does the Year 11 Cambridge book introduce oblique asymptotes using limits?

 

[cutemouse]

divide everything by highest power of x. take limit of x going to infinity.

:S But that would yield no limit if the degree of the leading term on the numerator is higher than the degree of the leading term on the denominator... (ie. when an oblique asymptote exists).

 

[jet]

What the syllabus states and what the authors decide to include in their textbooks are are not mutually inclusive in my opinion.

For example, Cambridge 3 unit teaches implicit differentiation which is not required in the 3 unit course, and is not stated in the syllabus.

Even then, in the Cambridge book, the authors acknowledge that Oblique asymptotes are not appropriate to be covered in depth and are only included for interest. It is on page 104 of the Year 11 book for those who would like to confirm this.

 

[shaon0]

I would still recommend 3U students to go through this because I've seen a few questions asked in assessments and trials of schools.

However, I haven't ever seen them in 4U exams.

lol haha. This is coming from the guy that told ppl to study for non-uniform circular motion.

What the syllabus states and what the authors decide to include in their textbooks are are not mutually inclusive in my opinion.

For example, Cambridge 3 unit teaches implicit differentiation which is not required in the 3 unit course, and is not stated in the syllabus.

Even then, in the Cambridge book, the authors acknowledge that Oblique asymptotes are not appropriate to be covered in depth and are only included for interest. It is on page 104 of the Year 11 book for those who would like to confirm this.

Yeah, in the extension questions and some intermediate questions of Cambridge yr11 and 12. The author introduces power series and other uni topics which aren't ever studied in high school.