Asymptotic Direction

[mathematics]

Please I am working with Functions , I want more explanations on What does an Asymptotic direction is : is it a new line like an oblique asymptote or is it just show the direction of the oblique asymptote -- What use can be benifit from the Asymptotic direction.

I need your help -- Please Forgive Me If I was un-clear. Thank you.

 

[tkhunny]

Asymptotic behavior comes in all shapes and sized. When dealing with RATIONAL functions, you need only observe the DEGREE of numerator and denominator to discern the situation. Usually, Asymptotes are grouped in just a few types, Horizotal, Vertical, Obilque, and "Higher Order". The first three are linear. If you had an example or to, and could show your workings, there are many who can provide more specific direction.

 

[mathematics]

Thanks for the reply you gave, I'm dealing with irrational functions.
See this : if limf(x) as x tends to \(\displaystyle \pm\infty\) = \(\displaystyle \pm\infty\)

The book says that there is 3 situations you must look for:

First: \(\displaystyle \lim\frac{f(x)}{x}\) as x tends to infinity =\(\displaystyle \pm\infty\) -->this means that y axis is the asymptotic direction of the graph..( what do they mean by this ?) I need some skectch examples please or websites.

Second: \(\displaystyle \lim\frac{f(x)}{x}\) as x tends to infinity = 0 --> Asymptotic direction is in the direction of x axis.

I need to understand this one to understand the third please.

 

[tkhunny]

First, Pick anything where f(x) is of degree 2 or greater.

Second, Pick anything where f(x) is of degree zero (0) - a constant.

 

[mathematics]

Thanks Tkhunny for you reply, I still dont get you . What I understood that thier is vertical , horizantal and oblique dealing with linear equations... So what to do with Asymptotic Direction-- One can say that it shows where the +infinity lies and the
-(mines) iinfinity lies ?. Is that true.

 

[tkhunny]

My first three impressions suggest that "asymptotic direction" has little to do with asymptotic behavior as encountered in your basic analytic geometry course. Your book MUST have provided a definition of "asymptotic behavior". Can you find it and print it here?

Further, what course are you in? Does it have a course number and title or are you wandering about on your own?

 

[Aladdin]

[\(\displaystyle \lim\frac{f(x)}{x} = a\)

\(\displaystyle b=\lim[f(x)-ax]\)

We may have;

\(\displaystyle \lim\frac{f(x)}{x}=a\) .....But \(\displaystyle b=\pm\infty\)

We say that it has an asymptotic direction in the direction of the line with slope a.

When : \(\displaystyle \lim\frac{f(x)}{x}=\pm\infty\)

We say that y-axis is an asymptotic direction of the graph.

 

[mathematics]

I'm dealing with irrational functions.

Given the function below.
\(\displaystyle f(x)=2x+1-2\sqrt{2x^2-x-1}\)

Q ) Discuss the variation and draw the graph :

What I've done : Domain= \(\displaystyle ]-\infty,-1/2]U[1,+\infty[\)

Endpoints: A(-1/2,0) , B(1,3)

Asymptotes:

Since the sign of 2 is positive , So we have two oblique asymptotes : y= ax+b :

\(\displaystyle a=\lim\frac{f(x)}{x}\) ,

and b= lim[f(x)-ax].

I dont know how to get them ... The books solution for this is :

y=2(1-sqrt2)x+1+sqrt3/2 , as x tends to +infinity.

y=2(1+sqrt2)x+1-sqrt3/2, as x tends to -infinity.

Please can anyone explain to me how this is done please.

I'll show the forward steps after I understand this one please.

Thank you for the help.