
An asymptote is a linear boundary that a curve gets very close to, but never reaches.^{[1]} Figure 1 is the graph of the function f(x) = 1/x. The vertical asymptote is the line x = 0. Notice that f(x) is undefined at x = 0. This means that f(x) gets very close to 0, but never reaches 0. So x = 0 is an asymptote of f(x) = 1/x. f(x) = 1/x also has a horizontal asymptote. f(x) can never be 1 because there is no value of x such that 1/x is defined. The horizontal asymptote of f(x) = 1/x is f(x) = 1. 

Figure 2 is a graph of the function f(x)=1/(x+1) What are the asymptotes of this function? Click here to see the answer.
The asymptotes of
f(x)=1/(x+1)
are
x=1
and
f(x)=0

Click on the points on the slider bars to change the figure. As you slide the bars, the equation for the function changes. This also changes the asymptotes. What is the relationship between the function and the asymptotes? 
Manipulative 1  Asymptote Created with GeoGebra. 
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