Page 43 - Calc for the Clueless

Oblique (Slanted Line) Asymptote

This occurs when the degree of the top is exactly 1 more than the bottom.

Example 10—

Degree of the top = 2; degree of the bottom = 1. Oblique asymptote.

We must, unfortunately, long divide the bottom into the top. If you know it, use synthetic division.

As x goes to infinity, the remainder 21/(x + 4) goes to 0. The oblique asymptote is y = x - 6.

Note 1

If the degree of the top is more than the bottom but not 1, there are no oblique asymptotes.

Note 2

At most there is one oblique asymptote or one horizontal asymptote, but not both. There might be neither.

Curve Sketching By the Pieces

Before we take a long example, we will examine each piece. When you understand each piece, the whole will
be easy.

Example 11—

The intercept is (4,0). We would like to know what the curve looks like near (4,0). Except at the point (4,0), we
do not care what the exact value is for y, which is necessary in an exact graph. In a sketch we are only interested
in the sign of the y values. We know f(4) = 0. f(3.9) = 3(3.9 - 4) = 0.000003. We don't care about its value. We
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only care that it is positive. Our notation will be f(4-) is positive. Similarly f(4 ) is positive. What must the
+
picture look like? At x = 4, the value is 0. To the left and to the right, f(x) is positive.

The picture looks like this where a = f(4 ) and b = f(4 )...
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Example 12—