How do you find the horizontal/slant asymptotes of a function?

Question

I know that to find the vertical asymptotes, you set the denominator equal to zero, but how do you find the horizontal or slant asymptotes?

For example: (2x-4)/(x^2-4)
The denominator is x^2-4, so that's (x+2)(x-2).
So the vert. asymptotes are x=2 and x=-2.
What about the horiz./slant asymptotes?

Answer 1

Let x go to infinity.

If x becomes really large, all smaller powers of x become much smaller than the greatest power, so from additions and subtractions you can remove all smaller powers. In your case,

(2x - 4) / (x^2 - 4) becomes 2x / x^2

Now apply the rules for multiplications and divisions of powers to simplify. Cancel all negative powers of x.

2x / x^2 becomes 2/x --> 0 so y = 0 is horizontal asymptote.

Slanted asymptote is when your result has the form a x + b.

Answer 2

Horizontal asympotets are lines of the form y=c where c is the limit of your function as x goes to infinity or negative infinity. Slant asymptotes are harder, so I will not try to explain everything here but will tell you about the most common situation. Suppose
f(x)=p(x)/q(x)
where p and q are polynomials, and degree of p=degree of q+1.
Ex: p(x)=x^3-2x+1, q(x)=x^2-4

Then you don't have horizontal asymptotes (the limits are +-infinity), but you have slant asymptotes. All you have to do is long division of polynomials with remainder, and then take the quotient.

Ex: x^3-2x+1= x(x^2-4)+ (2x+1)
Equation of the slant asymptote: y=x

Answer 3

You take limits of the function to examine behavior...

lim x-> infinity (2x-4)/(x^2-4) =
divide numerator and denominator by x^2
lim x-> infinity (2/x-4/x^2)/(1-4/x^2)
obviously lim x-> infinity a/x^n = 0
= (0-0)/(1-0) = 0

Analagous argument with lim x-> - infinity

It's been shown the function approaches the horizontal asymptote y=0 in both directions.

A slant asymptote is obtained by examining the limit of the derivative of the function as x approaches +/- infinity.

Answer 4

The way I see it asymtotes are number that drive the function to infinity. I don't know if there are horizontal Asymptotes. There are min and max which can be found by taking the derivitive and equaling it to zero.

Answer 5

2. C count number the type of signal alterations (from + to -). There are 4 so there are 4 or 2 effective actual recommendations. x^4 tells us the type of attainable recommendations. a number of those recommendations might want to be effective, detrimental, or complicated.

Answer 6

Careful. You can't set the denominator to zero. Thou shalt not divide by zero.