In the interval 0 </ x < 2pi, determine the equations of all asymptotes for the function y=tan x.
a) x=0
b) x=pi/2
c) x=0, x=pi
d) x=pi/2, 3pi/2
See if you can figure out what the values of tan(x) are at x=0,π/2, π and 3*pi;/2 to get an idea.
Even better, look up the graph of tan(x), noting that tan(-x)=-tan(x) (odd function) and tan(x) = tan(x+π).
See for example:
http://images.google.ca/imgres?imgurl=http://www.analyzemath.com/trigonometry/graph_tangent.gif&imgrefurl=http://www.analyzemath.com/trigonometry/properties.html&usg=__15mZqtEGVbBJpy38TrM4b8suxSM=&h=297&w=338&sz=5&hl=en&start=2&tbnid=0RO3fUwazvjljM:&tbnh=105&tbnw=119&prev=/images%3Fq%3Dtangent%2Bfunction%26gbv%3D2%26hl%3Den%26sa%3DG
To determine the equations of the asymptotes for the function y = tan(x) in the interval 0 < x < 2pi, we need to understand the behavior of the tangent function.
The tangent function has vertical asymptotes at values where the denominator of the fraction becomes zero. In the case of y = tan(x), the denominator is cos(x), which equals zero when x = pi/2 and 3pi/2. Therefore, the vertical asymptotes are at x = pi/2 and x = 3pi/2.
Next, we need to consider if there are any horizontal asymptotes. For the function y = tan(x), there are no horizontal asymptotes. The tangent function oscillates between positive infinity and negative infinity as x approaches the vertical asymptotes.
So the correct option is d) x = pi/2, 3pi/2. These are the equations of the vertical asymptotes for the function y = tan(x) in the interval 0 < x < 2pi.