Question 196162
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There are infinitely many asymptotes for each of
these.

Rules:

To find asymptotes for 

        Tangent and secant graphs

Set the argument (what the tangent or secant is of)
equal to {{{(2n+1)*pi/2}}} and solve for x.

Then let n = any integer, positive, negative, or zero.

        Cotangent and cosecant graphs

Set the argument (what the cotangent or cosecant is of)
equal to {{{n*pi}}} and solve for x.

Then let n = any integer, positive, negative, or zero.

[Note: the asymptotes are not affected by a coefficient
in front of the trig function]

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1. What are the asymptotes of {{{g(x)= cot(0.5x)}}}
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This is a cotangent graph, so set

{{{0.5x = n*pi}}}

Remove the decimal by multiplying both sides by 10

{{{5x = 10n*pi}}}

{{{x = (10n*pi)/5}}}

{{{x = 2n*pi}}}  

Let n = -3, -2, -1, 0, 1, 2, 3

{{{x = 2(-3)*pi}}},{{{x = 2(-2)*pi}}}, {{{x = 2(-1)*pi}}},{{{x = 2(0)*pi}}}, {{{x = 2(1)*pi}}}, {{{x = 2(2)*pi}}}, {{{x = 2(3)*pi}}}, etc., etc.,

This becomes:

{{{x = -6pi}}},{{{x = -4pi}}}, {{{x = -2pi}}},{{{x = 0}}}, {{{x = 2pi}}}, {{{x = 4pi}}}, {{{x = 6pi}}}, etc., etc.,

These are all asymptotes.

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2. What are the asymptotes of {{{g(x) = sec(2x)}}}
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This is a secant graph, so set

{{{2x = (2n+1)*pi/2}}}

{{{x = ((2n+1)*pi/2)/2}}}

{{{x = (2n+1)*pi/2}*(1/2)}}}

{{{x = (2n+1)*pi/4}}}

  Let n = -3, -2, -1, 0, 1, 2, 3

{{{x = (2(-3)+1)*pi/4}}},{{{x = (2(-2)+1)*pi/4}}},{{{x = (2(-1)+1)*pi/4}}},{{{x = (2(0)+1)*pi/4}}},{{{x = (2(1)+1)*pi/4}}},{{{x = (2(2)+1)*pi/4}}},{{{x = (2(3)+1)*pi/4}}}

This becomes:

{{{x = -5pi/4}}},{{{x = -3pi/4}}}, {{{x = -pi/4}}},{{{x = pi/4}}}, {{{x = 3pi/4}}}, {{{x = 5pi/4}}}, {{{x = 7pi/4}}}, etc., etc.,

These are all asymptotes.

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3. What are the asymptotes of {{{g(x) = (1/2)csc(x)}}}
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Note: The {{{1/2}}} in front does not affect the asymptotes,

This is a cosecant graph, so set

{{{x = n*pi}}}

Let n = -3, -2, -1, 0, 1, 2, 3

{{{x = -3pi}}},{{{x = -2pi}}}, {{{x = -pi}}},{{{x = 0}}}, {{{x = pi}}}, {{{x = 2pi}}}, x = 3pi, etc., etc.,

These are all asymptotes.

Edwin</pre>