## Vertical asymptoteA line is a
vertical asymptote of a function when or
An asymptote is a line that is not part of the graph, but one that the graph approaches closely. When the graph gets close to the vertical asymptote, it curves either upward or downward very steeply so that it looks almost vertical itself. Remember that the graph can get very close to the asymptote but can't touch it. The following figure illustrates some of the ways in wich the graph of a function may approach its vertical asymptote: As an example, let's see the graph of :
Theorem on Vertical Asymptotes of Rational FunctionsIf the real number a is a zero of the demoninator Q(x) of a rational function, then the graph of f(x)=P(x)/Q(x), where P(x) and Q(x) have no common factors, has the vertical asymptote x=a.
Find the vertical asymptotes for
Graphically: Find the vertical asymptotes for
Graphically: After studying rational functions may seem that all functions work the same. We make the mistake of thinking that the vertical asymptotes are found only in the points outside the domain. To avoid falling into this error, we see that we can find functions with vertical asymptotes at points belonging to the domain of the function. Consider the following piecewise function that has a vertical asymptote at x =2 and this point is part of its domain: Find the vertical asymptotes of <2\\x-1\;\;\;si\;x\ge2 align='absmiddle'>: <2\\x-1\;\;\;si\;x\ge2 align='absmiddle'> has a vertical asymptote at x=2, and x=2 is a point of its domain. Graphically:
Find the vertical asymptote(s) of the function
There are |