How do you find the Vertical, Horizontal, and Oblique Asymptote given #s(t) = t / sin t#?

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Answer 1 · 2018-01-20T13:11:28

Vertical asymptotes where #t = npi: n in ZZ, n!=0#

#s(t) = t/sint#

#s(t)# is undefined whereever #sint =0#

I.e Where #t = npi: forall n in ZZ#

Now consider the graph of #s(t)# below.

graph{x/sinx [-46.2, 46.33, -23.06, 23.16]}

It can be seen that #s(t)# has vertical asymptotes where #t = npi: n in ZZ, n!=0#

Also note #lim_(t->o) t/sint=1# can be seen on the graph above.

#s(t)# has no other asymptotes.