# What are the Asymptotes of (X-5)/(X^2+X-6)?

• theo12349
In summary: Homework Statement Draw graph of the following equationHomework Equations## \frac {X−5} {X^2+X−6}=y ##The Attempt at a Solutionmy problem is searching for the Horizontal asymptote from what I know (froum schaum's Outlines College Mathemathics 3rd Edition) the way to find the Horizontal asymptote is by limiting the equation near to ∞. and based on my calculation, the horizontal asymptote should be 0 but when you put X=5, the graph will show that it crosses (5,0)
theo12349
Good afternoon

1. Homework Statement

Draw graph of the following equation

## Homework Equations

$$\frac{X-5}{X^2+X-6} = y$$

## The Attempt at a Solution

my problem is searching for the vertical asymptote

from what I know the way to find the vertical asymptote is by limiting the equation near to ∞. and based on my calculation, the vertical asymptote should be 0

but when you put X=5, the graph will show that it crosses (5,0)

Last edited:
theo12349 said:
Good afternoon

1. Homework Statement

Draw graph of the following equation

X-5/(X2+X-6 = 0

## The Attempt at a Solution

my problem is searching for the vertical asymptote

from what I know the way to find the vertical asymptote is by limiting the equation near to ∞. and based on my calculation, the vertical asymptote should be 1

but when you put X=5, the graph will show that it crosses (5,1)

I have no idea what your calculation is or how it tells you that the vertical asymptote should be 1. Vertical asymptotes are where the function grows indefinitely large at nearby values. I think you need to review the definition of 'vertical asymptote'.

Dick said:
I have no idea what your calculation is or how it tells you that the vertical asymptote should be 1. Vertical asymptotes are where the function grows indefinitely large at nearby values. I think you need to review the definition of 'vertical asymptote'.

I do know that vertical asymptotes are where the function grows indefinitely large at nearby values. and it will reache the value of the asymptotes when X=∞

from what I read at Schaum's outline book, you can find the vertical asymptote by using ## \lim_{n\rightarrow +\infty} f(X) ##

I use this formula and get 0 as the answer

I did put the wrong value, sorry for my mistake. I am recalling it from my memory

anyway, when I insert X=5 it reaches the value 0

Last edited:
theo12349 said:
I do know that vertical asymptotes are where the function grows indefinitely large at nearby values. and it will reache the value of the asymptotes when X=∞

from what I read at Schaum's outline book, you can find the vertical asymptote by using ## \lim_{n\rightarrow +\infty} f(X) ##
No -- that's a horizontal asymptote.
theo12349 said:
I use this formula and get 0 as the answer

I did put the wrong value, sorry for my mistake. I am recalling it from my memory

anyway, when I insert X=5 it reaches the value 0
That would be the x-intercept.

theo12349 said:
I do know that vertical asymptotes are where the function grows indefinitely large at nearby values. and it will reache the value of the asymptotes when X=∞

from what I read at Schaum's outline book, you can find the vertical asymptote by using ## \lim_{n\rightarrow +\infty} f(X) ##

I use this formula and get 0 as the answer

I did put the wrong value, sorry for my mistake. I am recalling it from my memory

anyway, when I insert X=5 it reaches the value 0

I doubt very much that the Schaum's outline book tells you to take ## \lim_{n\rightarrow +\infty} f(X) ##, because that expression makes no sense at all, and Schaum's outlines are never so careless as that. There is a difference between horizontal and vertical asymptotes; do you know how to distinguish between them?

On the substance, you can easily factorise the denominator and that should hopefully immediately reveal to you a lot about the 'vertical asymptotes'.

For starters, look at ##y = \frac1x##. This has a vertical asymptote at x = 0, since it is undefined at this value and for very small x, 1/x is very large.
Your confusion earlier, as Mark pointed out was that you were using the definition for a horizontal asymptote instead of the one for vertical.
The horizontal asymptote of 1/x is y=0. Which the function never reaches but nears as x gets very large.
Note that vertical asymptotes are almost always defined by values of x where the function is undefined...you may have covered these in exercises about the domain of a function.

Mod note: The OP started a new thread, but I have merged the new one to the existing thread.

some of you might already seen this question in my previous post, but since there is a lot of mistakes that I made. I decided to make a new thread. I'm terribly sorry for the confusion before.

1. Homework Statement

Draw graph of the following equation

## Homework Equations

## \frac {X−5} {X^2+X−6}=y ##

## The Attempt at a Solution

my problem is searching for the Horizontal asymptote

from what I know (froum schaum's Outlines College Mathemathics 3rd Edition) the way to find the Horizontal asymptote is by limiting the equation near to ∞. and based on my calculation, the horizontal asymptote should be 0

but when you put X=5, the graph will show that it crosses (5,0)
[sorry for my english]

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Last edited by a moderator:
theo12349 said:
Mod note: The OP started a new thread, but I have merged the new one to the existing thread.

some of you might already seen this question in my previous post, but since there is a lot of mistakes that I made. I decided to make a new thread. I'm terribly sorry for the confusion before.

1. Homework Statement

Draw graph of the following equation

## Homework Equations

## \frac {X−5} {X^2+X−6}=y ##

## The Attempt at a Solution

my problem is searching for the Horizontal asymptote

from what I know (froum schaum's Outlines College Mathemathics 3rd Edition) the way to find the Horizontal asymptote is by limiting the equation near to ∞. and based on my calculation, the horizontal asymptote should be 0
Yes, that's correct. It is also useful to find the limit of this function as x approaches -∞. Some functions have different horizontal asymptotes as x → ∞ and as x → -∞.
theo12349 said:
but when you put X=5, the graph will show that it crosses (5,0)
The horizontal asymptote indicates behavior for very large values of the independent variable x, or for very negative values. The graph can still cross the x-axis at reasonably small values of x, such as x = 5.
theo12349 said:
[sorry for my english]

How does your graph look? There should be two vertical asymptotes and you should be able to determine where the horizontal asymptotes are for x going to positive and negative infinity. I mean, is the plot above or below the horizontal asymptotes in the limit?
You should be able to describe the plot generally by its behavior near these limiting and critical values.

## 1. What is an asymptote?

An asymptote is a line that a graph approaches but never touches. It can be either horizontal, vertical, or slanted.

## 2. How do you find the asymptotes of a rational function?

To find the asymptotes of a rational function, set the denominator equal to zero and solve for the value(s) of x. These values will be the vertical asymptotes. Then, use long division or synthetic division to simplify the function and determine the horizontal asymptote.

## 3. Are there any asymptotes for the function (X-5)/(X^2+X-6)?

Yes, there are two asymptotes for this function. The vertical asymptote is x = 2, and the horizontal asymptote is y = 0.

## 4. How do you graph a rational function with asymptotes?

To graph a rational function with asymptotes, plot the vertical asymptote(s) as a dotted line, plot points on either side of the asymptote(s) and connect them with a smooth curve. Then, plot the horizontal asymptote(s) as a dotted line and extend the graph to approach the asymptote(s) but never touch it.

## 5. Can a rational function have more than one vertical asymptote?

Yes, a rational function can have more than one vertical asymptote. The number of vertical asymptotes is equal to the degree of the polynomial in the denominator of the function.

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