f(x)=6/x+5

Question

Find the horizontal or oblique asymptote(s), if any,

Find the vertical asymptote(s) if any,

Describe their possible meanings

Answer 1

Hi Lauren!

 

If your equation is f(x) = __6__, I agree with Philip. In a quotient, a vertical asymptote always occurs where

                                        x + 5

the denominator = 0, so here that would be x = -5. The horizontal asymptote is at y = 0, thought which class you're in has an effect on your reasoning. If you're in Calculus, you can look at the limit as x approaches infinity, as Philip said. If not, there are three ways to approach it. First, you can try really high values of x (like x = 100, x = 1000, etc) and you can see that the y values are getting tiny. Second, you can look at a graph of the function and you'll see that the graph gets closer and closer to the y axis (x = 0). Third, you can consider the "behavior" of the graph. This guy will act like 1/x. If you know your basic graphs, this can help you as well.

 

Oblique asymptotes occur when the highest power of x on the top is exactly one more than the highest power of x on the bottom. That's not the case here, so there are no oblique asymptotes.

 

HOWEVER....

 

If your equation is as you wrote it, f(x) = _6_  + 5, you have a different beast. Before we can determine

                                                                  x

anything from this guy, we have to make it look like a complete quotient, which is what we're used to. To do that, we need a common denominator, which in this case is x. We have to multiply the 5 by x/x, then add the numerators:

 

f(x) = _6_  + _5x_ = _6 + 5x_

            x           x            x

 

This changes everything. A vertical asymptote again happens whenever the denominator = 0, so in this case, that's at x = 0. For the horizontal asymptote, we can use any of our methods.

 

1. The limit as x = 0 of f(x) is 5.
2. Trying high values of x will show that the y values are getting closer to 5.
3. Looking at the graph will show that there is an invisible barrier that the equation doesn't cross...a horizontal line at y = 5.
4. Looking at the behavior, we see that the highest power on the top and bottom are equal. in this case, we look at the coefficients of those terms that have the highest power of x. Here, that would be the 5 on top and a 1 on the bottom. This simplifies to 5... again showing a horizontal asymptote at y = 5.

 

Anyway you cut it, there is a horizontal asymptote at y = 5.

 

As before, for an oblique asymptote, the power on top has to be exactly one more than the power on bottom. That's not the case, so no oblique asymptote.

Answer 2

f(x) = 6/(x+5)

 

There will be a vertical asymptote whenever the denominator = 0, which occurs at x = -5.

 

f(x) --> 0 as x --> ±∞, so the horizontal asymptote is the x-axis (y=0)

 

For x>-5, f(x) is positive.  For x<-5, f(x) is negative