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