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

Question

Elijah Allen · Accepted Answer

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.

Serenity Johnson · Answer

undefined To find the vertical asymptotes of $ s(t) = \frac{t}{\sin(t)} $, identify the values of $ t $ where the denominator ($ \sin(t) $) equals zero. These values will result in vertical asymptotes.

The horizontal asymptote can be found by analyzing the behavior of the function as $ t $ approaches positive or negative infinity.

To find oblique asymptotes, perform polynomial long division to divide the numerator ($ t $) by the denominator ($ \sin(t) $). The quotient obtained from this division represents the equation of the oblique asymptote.

In summary:
- Vertical asymptotes occur where $ \sin(t) = 0 $.
- The horizontal asymptote can be determined by analyzing the limit of the function as $ t $ approaches positive or negative infinity.
- Oblique asymptotes can be found through polynomial long division of the numerator by the denominator.