Limit of Cosine

Question

lim as x approaches infinity Cos(x)

I'm pretty sure it doesn't exist, but my assignment doesn't like that answer

Answer 1

You are absolutely correct! In order for a limit to exist, it has to get closer and closer to one specific number. The graph of cosine goes back and forth between 1 and -1 forever, so there is not one specific number.

It's possible that your assignment just wants the answer in a different format? Try "DNE", "Limit does not exist" and "Undefined."

Answer 2

Cosine of an angle oscillates between positive and negative one never reaching a limit

just look at a graph, as x goes to infinity, cosx never converges

it never converges, at least not to any real number

but there is a paradox illustrated by Euler's Identity

e^ix = cosx +isinx

where when x=pi,

e^ip = cospi +isinpi = -1+0 = -1

that's well accepted as an identity, always true

but then take natural logs of both sides

ipi = ln(-1)

solve for pi which we know is about 3.141593...a real but irrational number

pi = ln(-1)/i

on the right side is a denominator which is imaginary and a numerator which takes the log of a negative number which has no real solution. log(-1) = ipi which is imaginary

pi = an imaginary number divided by an imaginary number = a real number

which leads to a question, is it possible the limit of cos(x) is an imaginary number?

probably not? but then it's not so certain

Euler is also known for "e" as the limit of (1+1/x)^x = e as x approaches infinity

he's the most prolific mathematician, writing more math than anyone. He also wrote more on limits, some less well known. Dig through it all and he may have written somewhere a little more about the limit of cos(x)

think of cosine values as like parallel lines extending infinitely out into space, that never intersect,so their is no convergence even as x approaches infinity. but is that true? what if space is curved like the earth's sphere where parallel longitude lines meet? non Euclidian geometry.