Finding Horizontal/ Slant/ Curvilinear Asymptotes Examples

Find any horizontal / slant / curvilinear asymptotes of the function

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We do long division:

Now we can rewrite the function f as

We know that the limit of  as x approaches ∞ or -∞ is 0.

As x gets very large or very negative,  is so small we don't need to bother with it. It's like when x is very large or very negative, f(x) is pretending to be 2. Since this is a constant, we have a horizontal asymptote at y = 2.

Find any horizontal / slant / curvilinear asymptotes of the function

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We start with long division:

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We can rewrite the function f as

When x gets very large or very negative,  is so small that we don't need to bother with it. f is pretending to be x + 1, which is a straight (but not horizontal) line. We've found a slant asymptote for the function f. Go us.

Find any horizontal / slant / curvilinear asymptotes of the function

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As always, we begin with long division:

Now we see that

As x gets very large or very negative,  gets so small it doesn't count for anything. For very large or very negative values of x, f(x) is pretending to be x² - x + 4.

Since this is not a straight line, horizontal or otherwise, we've found a curvilinear asymptote for the function f(x).

Find the horizontal asymptote of the function

The leading term of the numerator is 20x⁷ while the leading term of the denominator is 6x⁷. Dividing, we see there is a horizontal asymptote at 

Since the degree of the numerator was the same as the degree of the denominator, we were able to skip all that painful long division, and just take the ratio of the leading coefficients for our asymptote. Neat trick, huh?