Two functions f(x) g(x) are called asymptotic f(X) g(x)--1 as x--(infiniti sign). Find two polynomials which are asymptotic. (Hint : think about horizontal asympototes. how can you get a horizontal aymptote of 1?)

Question

Two functions f(x) g(x) are called asymptotic f(X) g(x)-->1 as x-->(infiniti sign). Find two polynomials which are asymptotic. (Hint : think about horizontal asympototes. how can you get a horizontal aymptote of 1?)

tkhunny · Answer

Answer that last question.

anonymous · Answer

what question?

tkhunny · Answer

how can you get a horizontal aymptote of 1?)

anonymous · Answer

im not sure

anonymous · Answer

?

tkhunny · Answer

Have you studied Rational Functions at all?

anonymous · Answer

a little bit , i understand a bit

tkhunny · Answer

Degree of Numerator Less Than Degree of Denominator
Horizontal Asymptote, y = 0

$y = \dfrac{1}{x+1}$

$y = \dfrac{x-4}{x^{2}-1}$

Degree of Numerator Equal To Degree of Denominator
Horizontal Asymptote, y = (Ratio of Leading Coefficients)

$y = \dfrac{2x}{x+1}$  --  y = 2/1 or just y = 2

$y = \dfrac{x^{2}-4}{3x^{2}-1}$  --  y = 1/3

Now, I ask again, how do you get an horizontal asymptote of y = 1?

anonymous · Answer

is it like x^2 , over x^2?

tkhunny · Answer

You have it!  Or (3x)/(3x) or (9x^4)/(9x^4).  As long as the Degree is the same and the leading coefficient is the same.  Excellent work.

anonymous · Answer

oh okay, so does that answer the question tho?

tkhunny · Answer

Of course, those are pretty boring.  You may wish to dress them up a bit  $$y = \dfrac{\sqrt{2}x^{2} - 5x + 11}{\sqrt{2}x^{2} + 3x - 7}$$

anonymous · Answer

where did the square root come from tho?

tkhunny · Answer

The question asks for two polynomials.  I just pulled them out of a hat.  ALL I care about it making the degree the same (2) and making the leading coefficient the same.  $\sqrt{2}$ may not be pretty, but as long as it's the same for both, we're good.

anonymous · Answer

oh so it can be any kind then?oh oh okay just to where we have the 1 coefficient?

tkhunny · Answer

Degree the same
Leading coefficient the same

Horizontal Asymptote: y = 1

Every time!

Nothing else matters for the location of the horizontal asymptote.

anonymous · Answer

oh okay, so all those words are just there to confuse me then?

tkhunny · Answer

"Find two polynomials which are asymptotic"  --  Pick any two you like.  You don't have to use mine.

anonymous · Answer

um 4x^3+2x^2+x/ 4x^3+x^2+x ?

tkhunny · Answer

Yes, but very bad notation: (4x^3+2x^2+x)/ (4x^3+x^2+x )

The parentheses are NOT optional.  Do you know why?

anonymous · Answer

yea, i coulndt think of another one . um not really

tkhunny · Answer

Order of Operations.

x + 1 / x + 2 = $x + \dfrac{1}{x} + 2$

(x + 1) / (x + 2) = $\dfrac{x+1}{x+2}$

It is a significant difference, wouldn't you say?

anonymous · Answer

yeah, i see the difference

tkhunny · Answer

Very good.  Be careful with that and move on to the next problem.

anonymous · Answer

okay thank you very much