Chapter 4 | Applications of Derivatives
419
2 +2 x 4 x 3 −5 x +7
b. f ( x ) = 3 x
( Note: The degree of numerator is less than the degree of the denominator.)
2 +4 x
c. f ( x ) = 3 x
( Note: The degree of numerator is greater than the degree of the denominator.)
x +2
Solution a. The highest power of x in the denominator is x . Therefore, dividing the numerator and denominator by x and applying the algebraic limit laws, we see that lim x →±∞ = lim x →±∞ 3−1/ x 2+5/ x 3 x −1 2 x +5
lim x →±∞ (3−1/ lim x →±∞ (2+5/
x ) x )
=
lim x →±∞ 3− lim x →±∞ 1/ x lim x →±∞ 2+ lim x →±∞ 5/ x
=
3 2 .
= 3−0 2+0 =
f ( x ) = 3
Since lim
we know that y = 3 2
is a horizontal asymptote for this function as shown in
2 ,
x →±∞
the following graph.
Figure 4.53 The graph of this rational function approaches a horizontal asymptote as x →±∞. b. Since the largest power of x appearing in the denominator is x 3 , divide the numerator and denominator by x 3 . After doing so and applying algebraic limit laws, we obtain lim x →±∞ 3 x 2 +2 x 4 x 3 −5 x +7 = lim x →±∞ 3/ x +2/ x 2 4−5/ x 2 +7/ x 3 = 3 ( 0 ) +2 ( 0 ) 4−5 ( 0 ) +7 ( 0 ) =0. Therefore f has a horizontal asymptote of y =0 as shown in the following graph.
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