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.
