Limit as x -> infinity of a sine/cosine graph.
1. The problem statement, all variables and given/known data
Lim [2 + 3x + sin(x)] / [x + 2cos(x)]
2. Relevant equations
3. The attempt at a solution
My roommate asked me to help him solve this homework question, at first glance I noted the derivative to be:
[3 + cos(x)] / [1 - 2sin(x)]
Now, the next step I'm assuming would be to plug infinity in for x. The answer in the book says that this should reduce the equation to 3. This would mean that by substituting infinity for x, the cos(x)/2sin(x) has to reduce to 1. But if x is approaching infinity, wouldn't a sin or cos wave just go back and forth between -1 and 1 forever? How, mathematically speaking, is it justifiable to say that they'd reduce to 1?