How do you find #lim_(xtooo)(Cos(x)/x)#?

3 Answers
Mar 28, 2017

Because the upper bound of cos(x) is 1 and the lower bound is -1:

#lim_(xtooo)-1/x<= lim_(xtooo)cos(x)/x <= lim_(xtooo)1/x#

We know the limits of the two bounds to be 0:

#0<= lim_(xtooo)cos(x)/x <= 0#

#:.#

#lim_(xtooo)cos(x)/x = 0#

Mar 28, 2017

Answer:

It should tend to zero.

Explanation:

I would say that as cos(x) oscilates between #1 and -1# then #x# will grow so much that the fraction will become very small tending to become zero.
This is somewhat supported by the graph of our function:

graph{(cos(x))/x [-10, 10, -5, 5]}

Because #-1 <= cos x <= 1# , we have that

#-1/x<=cosx/x<=1/x#

Hence

#lim_(x->oo) (-1/x)<=lim_(x->oo) cosx/x<=lim_(x->oo) 1/x#

#0<=lim_(x->oo) cosx/x<=0#

Finally #lim_(x->oo) cosx/x=0#