Chapter 4 | Applications of Derivatives
411
Figure 4.44 This function crosses its horizontal asymptote multiple times.
−1 ( x ) and lim
−1 ( x ), we first consider the graph of y = tan( x ) over the
c. To evaluate lim x
→∞ tan
x →−∞ tan
interval (− π /2, π /2) as shown in the following graph.
Figure 4.45 The graph of tan x has vertical asymptotes at x =± π 2
Since
lim x →( π /2) −
tan x =∞,
it follows that
−1 ( x ) = π
lim x →∞ tan
2 .
Similarly, since
lim x →(–π/2) +
tan x =−∞,
it follows that
−1 ( x ) = − π
lim x →−∞ tan
2 .
As a result, y = π 2
and y = − π 2
are horizontal asymptotes of f ( x ) = tan −1 ( x ) as shown in the following
graph.
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