Which type of asymptote, when it occurs, describes the behavior of a graph when x is close to some number?

A slant or oblique asymptote.

Explanation:
When a function is a rational function whose denominator is one degree lower than the numerator, a slant or oblique asymptote occurs when the function is graphed.

Example:
Consider the rational function
y = (2x² + 2x + 3)/(x + 1)
The degree of the numerator is 2, and the degree of the denominator is 1 so that the degree of the numerator is one higher than that of the denominator.

After long division, obtain
                    2x
       -----------------------
x+1  |  2x²  + 2x  + 3
          2x²  + 2x
         -------------------
                             3

That is,
(2x²+2x+3)/(x+1) = 2x + 3/(x+1)
As x -> ∞, 3/(x+1) -> 0, and the curve behaves as the straight line y = 2x.
The curve, therefore, approaches the straight line with a slope of 2.
For this reason, it is called a slant or oblique asymptote.

The graph of this function is shown below to illustrate the concept.

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