Horizontal Asymptotes

[jesusphreek82]

I dont understand how to find horizontal Asymptotes .... can anyone help me with this problem??

y= (x-3)/(x-2)

this is due tomorrow.... I really need help! I would appreciate it :lol:

 

[soroban]

Hello, jesusphreek82!

Find horizontal asymptote: \(\displaystyle \,\L y\:=\:\frac{x\,-\,3}{x\,-\,2}\)

We want to know what happens to $\displaystyle y$ as $\displaystyle x$ get very very large
. . . either positively or negatively.

That is, does $\displaystyle \lim_{x\to\infty} y$ have a finite value?

$\displaystyle \;\;$Does $\displaystyle \;\lim_{x\to\infty}\left(\frac{x\,-\,3}{x\,+\,3}\right)$ have a limit?


"Eyeballing" it, it seems to go to $\displaystyle \frac{\infty}{\infty}$ . . . which is no help.


Here's the "trick": divide top and bottom by $\displaystyle x:$ **

\(\displaystyle \L\;\;\lim_{x\to\infty}\left(\frac{\frac{x}{x}\,-\,\frac{3}{x}}{\frac{x}{x}\,+\,\frac{3}{x}}\right) \;=\;\lim_{x\to\infty}\left(\frac{1\,-\,\frac{3}{x}}{1\,+\,\frac{3}{x}}\right) \;=\;\frac{1\,-\,0}{1\,+\,0}\;=\;1\)


We have shown that, as $\displaystyle x\to\infty,\;y$ approaches $\displaystyle 1.$

Therefore, the horizontal asymptote is: $\displaystyle \,y\,=\,1$

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** $\displaystyle \;$Rule

Divide top and bottom by the highest power of $\displaystyle x$ in the denominator