# How do you find the vertical, horizontal and slant asymptotes of: # f(x) = x/ sinx#?

##### 1 Answer

It has a hole at

#### Explanation:

Vertical asymptotes will occur where the denominator is zero and the numerator non-zero.

#sin x = 0# if and only if#x = n pi# for some#n in ZZ#

Hence

#lim_(x->0) x/(sin x) = 1#

#f(2n pi+pi/2) = 2n pi+pi/2 -> +oo# as#n -> +oo#

#f(2n pi+pi/2) = 2n pi+pi/2 -> -oo# as#n -> -oo#

#f(2n pi+(3pi)/2) = -(2n pi+(3pi)/2) -> -oo# as#n -> +oo#

#f(2n pi+(3pi)/2) = -(2n pi+(3pi)/2) -> +oo# as#n -> -oo#