Question

Explain why lim x->∞ (cos(x)) does not exist, but lim x->∞ ((cos(x))/(x)) does exist.

Answer 1

To have a defined limit, a function has to approach a value. When happens when lim x-> inf cos x?

Answer 2

it oscillates...

Answer 3

exactly, between 1 and -1. So it never tends toward a value. But what if there's an x on the bottom?

Answer 4

the limit equals 0?

Answer 5

yah! Even though cos x oscillates between 1 and -1, both of those values are forced to tend towards zero because of the 1/x.

Answer 6

okay, thank you :)

Answer 7

welcome :) That approach is called the squeeze theorem.