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Calculus AB: Applications of the Derivative

Vertical and Horizontal Asymptotes

Problems for "Using the Second Derivative to Analyze Functions"

Problems for "Vertical and Horizontal Asymptotes"

Vertical Asymptotes

A vertical asymptote occurs at x = c when the following are all true

    1) f (c) is undefined  
    2) f (x) = ∞ or - ∞  
    3) f (x) = ∞ or - ∞  

Taken together, #2 and #3 mean that f "grows without bound" as it approaches x = c . This happens most often with a rational function at a value of x that leads to a denominator of zero. For example, consider f (x) = . f (x) is undefined at x = - 1 .

    1) f (x) is undefined at x = - 1  
    2)  = - ∞  
    3)  = + ∞  

Thus, x = - 1 is a vertical asymptote of f , graphed below:

Figure %: f (x) = has a vertical asymptote at x = - 1

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches, but never touches as x approaches negative or positive infinity. If f (x) = L or f (x) = L , then the line y = L is a horiztonal asymptote of the function f. For example, consider the function f (x) = . This function has a horizontal asymptote at y = 2 on both the left and the right ends of the graph:

Figure %: f (x) = . Has a horizontal asymptote at y = 2

Note that a function may cross its horizontal asymptote near the origin, but it cannot cross it as x approaches infinity.

Intuitively, we can see that y = 2 is a horizontal asymptote of f because as x approaches infinity, f (x) = behaves more and more like f (x) = , which is the same as f (x) = 2 . Although f behaves more and more like this, it never actually becomes this function, so y = 2 is approached but not reached.

Intuition can usually lead to the right answer with these problems, but the following is a more methodical way of calculating limits at infinity.

Evaluating Limits at Infinity

In order to find horizontal asymptotes, we must evaluate limits as x approaches infinity. To evaluate the limits of rational functions at infinity, first divide each of the terms in the numerator and the denominator by the highest. For example, to evaluate


first divide each of the terms in the numerator and denominator by the highest power of x present in the function. In this case, that is x 3.


then evaluate the individual limits using the following rule: if r is a rational number greater than zero such that x r is defined for all x , then

= 0    

Applying this rule in this case leads to the following:


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