We want all of the asymptotes: vertical, horizontal, and slant. None of them will escape us. We'll start the search by checking for vertical asymptotes.
We see that the denominator will equal zero when . There's no way to factor 4x to cancel that out, so this is a vertical asymptote.
Now, what about horizontal and slant asymptotes? The degrees of the top and bottom polynomials are equal. That gives us a horizontal asymptote, at .
That's all the asymptotes. We'd stick them in our trophy case, but they won't fit because they extend out to infinity.
Example 2
Find all the asymptotes of .
We take a quick glance at the degrees of each polynomial, and notice that the bottom one is larger. That means we have a horizontal asymptote at y = 0. Nice.
Let's factor the denominator to find any vertical asymptotes.
We can't cancel them out, so there are vertical asymptotes of x = -1 and -3.
Example 3
Find all the asymptotes of .
First we'll check for vertical asymptotes and holes. F-f-factor that function:
Nothing is knocked out by factoring, so we have two vertical asymptotes, at x = -1 and 1.
Next comes the search for a horizontal or slant asymptote. The top degree is larger than the bottom degree, so we have some kind of slant asymptote. Show us that old time long division:
We drop the remainder like a hot potato, leaving us with a slant asymptote of y = 3x.
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. There's no way to factor 4x to cancel that out, so this is a vertical asymptote.
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