Rational functions and asymptotes

Dividing by a tiny number gives a huge result. So where a denominator hits zero, the graph shoots up or down, and that makes a vertical asymptote.

y = \frac{1}{x - 2} + 1

has a wall at

x = 2

and settles toward

y = 1

Degrees	Horizontal asymptote
Top smaller	$y = 0$
Same	Ratio of leading coefficients
Top larger	None

Work through it

The denominator is zero at 4. $x = 4$

The degrees match, so use $\frac{3}{1}$ . $y = 3$

Factor and cancel. $y = x + 2$

The cancelled factor leaves a hole, not a wall.Hole at $(2, 4)$

Calling every zero of the denominator an asymptote.

If the factor cancels, it's a hole.

The short version
Vertical asymptote: denominator zero, after cancelling. Horizontal: compare the top and bottom degrees.

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