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**Unformatted text preview: **ction, we see that J (x) = tan(x) is undeﬁned at
π
x = π and x = 32 , in accordance with our ﬁndings in Section 10.3.1. As x → π − , sin(x) → 1−
2
2
sin(
and cos(x) → 0+ , so that tan(x) = cos(x) → ∞ producing a vertical asymptote at x = π . Using a
2
x)
− + π
π
similar analysis, we get that as x → π + , tan(x) → −∞, as x → 32 , tan(x) → ∞, and as x → 32 ,
2
tan(x) → −∞. Plotting this information and performing the usual ‘copy and paste’ produces:
y x
0 tan(x) (x, tan(x))
0
(0, 0)
π
4,1 π
4
π
2
3π
4 undeﬁned π 0 (π, 0) 5π
4
3π
2
7π
4 1 5π
4 ,1 undeﬁned 2π 0 1
−1 −1 1
3π
4 , −1 π
4 π
2 3π
4 π 5π
4 3π
2 7π
4 2π −1 7π
4 , −1 (2π, 0)
The graph of y = tan(x) over [0, 2π ].
y x The graph of y = tan(x). x 10.5 Graphs of the Trigonometric Functions 687 From the graph, it appears as if the tangent function is periodic with period π . To prove that this
is the case, we appeal to the sum formula for tangents. We have: tan(x + π ) = tan(x) + tan(π )
tan(x) + 0
=
= tan(x),
1 − tan...

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