Calculus AB : Connect Infinite Limits and Vertical/Horizontal Asymptotes
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Example Question #21 : Connect Infinite Limits And Vertical/Horizontal Asymptotes
Which of the four following functions is probably depicted in the sample data above?
Notice that as x increases, the points of data graphed appears to level out, flattening towards a certain value. This value is what is known as a horizontal asymptote. An asymptote is a value that a function approaches, but never actually reaches. Think of a horizontal asymptote as a limit of a function as x approaches infinity. In such a case, as x approaches infinity, any constants added or subtracted in the numerator and denominator become irrelevant. What matters is the power of x in the denominator and the numerator; if those are the same, then to coefficients define the asymptote. We see that the function flattens towards:
This matches the ratio of coefficients for the function:
Example Question #22 : Connect Infinite Limits And Vertical/Horizontal Asymptotes
Decide which of the following functions, most likely represents the sample data above.
Observation of the data points shows that there is a sharp increase and decrease on either side of a particular x-value. This type of behavior is observed when there is a vertical asymptote. An asymptote is a value that a function may approach, but will never actually attain. In the case of vertical asymptotes, this behavior occurs if the function approaches infinity for a given x-value, often when a zero value appears in a denominator. Noting this rule, the above function has a zero in the denominator at a definite point centered approximately around:
We find a zero denominator for the function:
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