Question

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

Answer 1

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`

Answer 2

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]}

Answer 3

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`