Prove that when x tends to infinity, the limit of " (cot x)/x " does not exist.

To show the limit of (cot x)/x as x-->infinity
.

Solution

(cotx)/x =
(cosx/sinx)/x = cosx/(xsinx).

Therefore Lt (cotx)/x =
Lt.(cosx)/(xsinx) as x-->infinity.Now we consider 3 cases (i) x = npi , (ii) x =
n(pi + a) and (iii) x = n(pi - a) where 0< a < pi, and n is a positive
integer.

Case (i) x = npi , cosnpi = 1 or -1 , and  sinnpi
= 0. But (cos npi)/(sin npi) is indeterminate . So at x= npi , cotnpi is indeterminate.
So at x= npi , Lt (cot npi)/npi is also indeterminate and does not exist .  Both right
limit  and  left limit are also indeterminates. So at x = npi , Lt n--> infinity
 (cot npi)/(npi) does ot
exist.

case(ii)

At x =  npi+ a
,  lt x--> inf  (cotx)/x = Lt n--> inf {cot(npi+a)/(npi+a) =
{cos(npi+a)/sin(pi+a) }/(npi+a) = (a finite number)/infinity = 0 for all 0 <  a
< pi.

case(iii) x=
npi-a.

Lt x--> inf (cotx)/x = Lt n-->inf
cot(npi-a)/(npi-a) = (A finite number)/ (infinity) = 0 for all 0 < a <
pi.

Therefore Lt x--> infinity (cotx)/x  does not
approach a unique limit.

Hope this
helps.