How can you tell when an equation has an oblique asymptote? (1 Viewer)

trolol101 · Jan 23, 2014

Something something about powers.

Carrotsticks · Jan 23, 2014

trolol101 said:
Something something about powers.

$\\ $Suppose you have an irreducible rational function in the form $ f(x)=\frac{A(x)}{B(x)} $ where $ A(x) $ and $ B(x) $ are polynomials.$ \\\\ $If $ \deg A > \deg B $, then $ f(x) = C(x) + \frac{R(x)}{B(x)} $ where $ \deg R < \deg B. $ The polynomial $ C(x) $ is your asymptote because as $ x \rightarrow \infty , \frac{R(x)}{B(x)} \rightarrow 0 $ so $ f(x) \rightarrow C(x). \\\\ $So say you have $ f(x) = \frac{x^2+1}{x+1}. $ We know that there will be an asymptote because the degree of the numerator is greater than the degree of the denominator$.$

QZP · Jan 23, 2014

^ Adding on, I believe it is only strictly correct to refer to asymptotes as straight lines. Thus you have an oblique asymptote when deg (A) - deg (B) = 1. Otherwise, the function is asymptotic to y = x^2, say.

Drongoski · Jan 23, 2014

Yep, qzp. When C(x) is linear.

Carrotsticks · Jan 23, 2014

QZP said:
^ Adding on, I believe it is only strictly correct to refer to asymptotes as straight lines. Thus you have an oblique asymptote when deg (A) - deg (B) = 1. Otherwise, the function is asymptotic to y = x^2, say.

EDIT: Correct, thank you. I mis-took 'oblique' to be 'non-horizontal asymptotes'.

However, I would like to point out that asymptotes can be curves, sometimes called 'curvilinear asymptotes'.

But oblique asymptotes must be in the form y=mx+b, where m is non-zero.

trolol101 · Jan 23, 2014

Carrotsticks said:
$\\ $Suppose you have an irreducible rational function in the form $ f(x)=\frac{A(x)}{B(x)} $ where $ A(x) $ and $ B(x) $ are polynomials.$ \\\\ $If $ \deg A > \deg B $, then $ f(x) = C(x) + \frac{R(x)}{B(x)} $ where $ \deg R < \deg B. $ The polynomial $ C(x) $ is your asymptote because as $ x \rightarrow \infty , \frac{R(x)}{B(x)} \rightarrow 0 $ so $ f(x) \rightarrow C(x). \\\\ $So say you have $ f(x) = \frac{x^2+1}{x+1}. $ We know that there will be an asymptote because the degree of the numerator is greater than the degree of the denominator$.$

I don't understand that second paragraph...

asianese · Jan 23, 2014

deg: degree - i.e. the highest power of the polynomial.

e.g. 2x^4 + 4x has degree 4, since the highest power is 4.

braintic · Jan 24, 2014

QZP said:
^ Adding on, I believe it is only strictly correct to refer to asymptotes as straight lines. Thus you have an oblique asymptote when deg (A) - deg (B) = 1. Otherwise, the function is asymptotic to y = x^2, say.

Asymptotes don't have to be straight lines.

Wolfram definition: An asymptote is a line or curve that approaches a given curve arbitrarily closely.

It IS correct to say that y=x^2 is an asymptote of y=x^2 + 1/x.

How can you tell when an equation has an oblique asymptote? (1 Viewer)

trolol101

New Member

Carrotsticks

Retired

QZP

Well-Known Member

Drongoski

Well-Known Member

Carrotsticks

Retired

trolol101

New Member

asianese

Σ

braintic

Well-Known Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)