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Find the vertical asymptotes and removable discontinuities of the graph of f ( x ) = x 2 25 x 3 6 x 2 + 5 x .

Removable discontinuity at x = 5. Vertical asymptotes: x = 0 ,   x = 1.

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Identifying horizontal asymptotes of rational functions

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.

There are three distinct outcomes when checking for horizontal asymptotes:

Case 1: If the degree of the denominator>degree of the numerator, there is a horizontal asymptote    at y = 0.

Example:  f ( x ) = 4 x + 2 x 2 + 4 x 5

In this case, the end behavior is f ( x ) 4 x x 2 = 4 x . This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function g ( x ) = 4 x , and the outputs will approach zero, resulting in a horizontal asymptote at y = 0. See [link] . Note that this graph crosses the horizontal asymptote.

Graph of f(x)=(4x+2)/(x^2+4x-5) with its vertical asymptotes at x=-5 and x=1 and its horizontal asymptote at y=0.
Horizontal Asymptote y = 0 when f ( x ) = p ( x ) q ( x ) , q ( x ) 0 where degree of p < d e g r e e   o f   q .

Case 2: If the degree of the denominator<degree of the numerator by one, we get a slant asymptote.

Example:  f ( x ) = 3 x 2 2 x + 1 x 1

In this case, the end behavior is f ( x ) 3 x 2 x = 3 x . This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function g ( x ) = 3 x . As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of g ( x ) = 3 x looks like a diagonal line, and since f will behave similarly to g , it will approach a line close to y = 3 x . This line is a slant asymptote.

To find the equation of the slant asymptote, divide 3 x 2 2 x + 1 x 1 . The quotient is 3 x + 1 , and the remainder is 2. The slant asymptote is the graph of the line g ( x ) = 3 x + 1. See [link] .

Graph of f(x)=(3x^2-2x+1)/(x-1) with its vertical asymptote at x=1 and a slant asymptote aty=3x+1.
Slant Asymptote when f ( x ) = p ( x ) q ( x ) , q ( x ) 0 where degree of p > degree of  q by 1 .

Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at y = a n b n , where a n and b n are the leading coefficients of p ( x ) and q ( x ) for f ( x ) = p ( x ) q ( x ) , q ( x ) 0.

Example:  f ( x ) = 3 x 2 + 2 x 2 + 4 x 5

In this case, the end behavior is f ( x ) 3 x 2 x 2 = 3. This tells us that as the inputs grow large, this function will behave like the function g ( x ) = 3 , which is a horizontal line. As x ± , f ( x ) 3 , resulting in a horizontal asymptote at y = 3. See [link] . Note that this graph crosses the horizontal asymptote.

Graph of f(x)=(3x^2+2)/(x^2+4x-5) with its vertical asymptotes at x=-5 and x=1 and its horizontal asymptote at y=3.
Horizontal Asymptote when f ( x ) = p ( x ) q ( x ) , q ( x ) 0 where degree of  p = degree of  q .

Notice that, while the graph of a rational function will never cross a vertical asymptote    , the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.

Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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