Thread: Asymptotes - Help f(x)=x²-5x+7/(x-2)

f(x)=x²-5x+7/(x-2)

It asks me to give the vertical, horizontal and slant asymptotes if there are any.

This is what I have so far:

vertical asymptote: x cannot equal 2

I am now trying to find the slant asymptote by synthetic (long division) because x²-5x+7 does not factor out.


I am really stuck so may someone show me how to do this problem? I still need to find the slant and horizontal asymptotes.

Originally Posted by agentlopez

f(x)=x²-5x+7/(x-2)

It asks me to give the vertical, horizontal and slant asymptotes if there are any.

This is what I have so far:

vertical asymptote: x cannot equal 2

I am now trying to find the slant asymptote by synthetic (long division) because x²-5x+7 does not factor out.


I am really stuck so may someone show me how to do this problem? I still need to find the slant and horizontal asymptotes.

The vertical asymptote is not "x cannot equal 2" it is the line x = 2.



So the slant asymptote is .

For horizontal asymptotes we find the limit of the function as x goes to plus or minus infinity. You can see for yourself that this function blows up/down at positive/negative infinity. Or you can simply use the fact that a function with a slant asymptote will never have a horizontal asymptote.

-Dan

Originally Posted by topsquark

The vertical asymptote is not "x cannot equal 2" it is the line x = 2.



So the slant asymptote is .

For horizontal asymptotes we find the limit of the function as x goes to plus or minus infinity. You can see for yourself that this function blows up/down at positive/negative infinity. Or you can simply use the fact that a function with a slant asymptote will never have a horizontal asymptote.

-Dan

Can you show me how you got and what would I get as the horizontal asymptote? I know you used synthetic division, but I need help on how to get to and what would be my horizontal asymptote?

You don't need synthetic division:



For horizontal asymptotes, evaluate

&

Originally Posted by Krizalid

You don't need synthetic division:



For horizontal asymptotes, evaluate

&

Now one has ever showed me that you can just split a number and do that. Something new I learned^^. I still do not understand the horizontal asymptote that you are explaining to me. I need someone to be more detailed with this.

Read carefuly these lines:

Originally Posted by topsquark

So the slant asymptote is .

For horizontal asymptotes we find the limit of the function as x goes to plus or minus infinity. You can see for yourself that this function blows up/down at positive/negative infinity. Or you can simply use the fact that a function with a slant asymptote will never have a horizontal asymptote.

--

Now, if you evaluate as or does that make sense?

Originally Posted by Krizalid

Read carefuly these lines:



--

Now, if you evaluate as or does that make sense?

^^Not yet^^

Maybe I was not specific enough when I asked to be more descriptive. I am a beginner at this stuff and I realise that the best way to learn is to figure it out for yourself, but I am really stuck and need this explained 'THOROUGHLY'. Nothing is 'clicking' for me even when you basically just reworded it above. I need this explained as if this is something completely new to me. I really need this so please help me understand.


What your telling me is all slant asymptotes do not have a horizontal asymptote? I thought you needed the horizontal and vertical asymptotes for when there is a slant asymptote.

Think of it this way...

As ,

This is because as x goes to infinity, becomes smaller and smaller, eventually becoming practically non-existent. But the part keeps rising steadily as does, so slowly converges upon .

You don't need synthetic division:

How did you know to split the 7 into 6+1?

When I saw Dan's algebra, I was wondering how I could split the original ratio into two sums.

Then clearly we need to find something on the top which yields I thought in 'cause the first two terms are exactly & so from here, we're done, 'cause the third term is Since on the numerator we have the conclusion follows quickly.

--

This does not require much effort, it's just a way to avoid the looong division

Originally Posted by DivideBy0

Think of it this way...

As ,

This is because as x goes to infinity, becomes smaller and smaller, eventually becoming practically non-existent. But the part keeps rising steadily as does, so slowly converges upon .

Does that mean the horizontal asymptote is 2?

x=2
y=2

f(x)=x-3

Is that right?

Originally Posted by agentlopez

Does that mean the horizontal asymptote is 2?

x=2
y=2

f(x)=x-3

Is that right?

Yes, there is a horizontal asymptote at x = 2. Try graphing the function.