Asymptotes: If the graph of y = f(x) gets closer and closer to the line
y = L or x = a, then these lines are asymptotes.
Definition: If f(x) = L, then y = L is a Horizontal Asymptote.
y = 1 is a horizontal asymptote for each graph below
Note: To find the horizontal asymptotes for f(x), take the limit as of the given funtion. If this limit is a number, L, then y = L is a horizontal asymptote for the graph.
!!An abbreviated limit notation will be used in this review!!
|as||"y approaches 4 as x approaches 1" |
and means limit of the y values, as x goes to 1, is 4
|"x increases without bounds" and we say "x goes to infinity"|
|"x decreases without bounds"|
|"x approaches 5 from the right"|
x only has values greater than 5
|"x approaches 5 from the left"|
x less than 5
Definition: If as or as
then x = a is a Vertical Asymptote. In other words, the y values increase or decrease without bound as the x values get closer to the line x = a.
x = 2 is a Vertical Asymptote for each of the graphs below.
Note: To find the vertical asymptote, reduce f(x) then if x = a makes the denominator zero (and not the numerator), x = a is a vertical asymptote for the graph. The graph will NEVER cross a vertical asymptote. The function is undefined there.
Given: find the vertical and horizontal asymptotes. Then, sketch the graph.
To find the horizontal asymptotes, take the limit as
To find the vertical asymptotes, factor demonimator (after reducing if possible)
|denominator = 0 (numerator not 0)|
when x = 3 or x = -3
To sketch by hand:
Now you may plot more points or you can investigate the behavior of y = f(x) as the x values get close to the vertical asymptotes. If then the y values become larger and larger positive and the graph goes up. If then the y values become more and more negative and the graph goes down. The graph never crosses a vertical asymptote.
|Using the points above, sketch information
about the horizontal asymptotes, as, , and as , (behavior at left and right ends)
|Usually the outside branches of the graph approach thevertical asymptotes, or plot the y values for x close to the vertical asymptotes like x = 3.1, -3.1|
|In the middle of the graph, plot points inside the asymptotes, like x = 2.9, -2.9, or check to see if the graph crosses the horizontal asymptotes by setting y = -2 and see if you can solve for x.|