MHB What Are the Steps to Find the Horizontal Asymptote of f(x) = (x^2 - 9)/(x - 3)?

[mathdad]

Find the horizontal asymptote of f(x) = (x^2 - 9)/(x - 3). I need the steps not the solution.

 

[skeeter]

The given function has no horizontal asymptote

 

[tkhunny]

Find the horizontal asymptote of f(x) = (x^2 - 9)/(x - 3). I need the steps not the solution.

Standard Rule: Numerator and Denominator have the same "Degree". THAT will get you an Horizontal Asymptote.

 

[mathdad]

The given function has no horizontal asymptote

Good to know but why?

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Standard Rule: Numerator and Denominator have the same "Degree". THAT will get you an Horizontal Asymptote.

By "Degree" you mean POWER or EXPONENT, right?

What about y = (x^2 + 3)/(x^2 + 5)?

Can we say the horizontal asymptote is y = 1?

The coefficient of x^2 is 1 for the numerator and denominator.

y = (x^2)/(x^2)

y = 1

True?

 

[MarkFL]

Observe that:

$$\frac{x^2+3}{x^2+5}=\frac{x^2+5-2}{x^2+5}=1-\frac{2}{x^2+5}$$

As $x\to\pm\infty$, the rational term vanishes (gets smaller and smaller), and the entire expression therefore approaches 1. :)

In general, you are correct...it can be shown that:

$$\lim_{x\to\pm\infty}\left(\frac{\sum\limits_{k=0}^n\left(a_kx^k\right)}{\sum\limits_{k=0}^n\left(b_kx^k\right)}\right)=\frac{a_n}{b_n}$$

 

[mathdad]

Good to know.