Limits dealing with infinity!

[fdragon]

Ok here's the Problem....

use graphs and tables to find
......(a) Lim f(x)
x --->Positive infinity

......(b) Lim f(x)
x ----> Neg inf

......(c) Identify all horizontal asymptotes


1. f(x) = cos (1/x)

2. f(x) = sin 2x/x

Ok so I'm looking at my teacher's notes and when I was in class they kinda made sense but I don't even know where to start...! I put number 1 in my calculator and the graph is all messed up. In number one the table said error at 0, so I'm guessing maybe that is the limit of the graph, if it even exists.
Thanks!!

 

[tkhunny]

The first one should not be confusing you. What is cos(0)? Now, think about the limit of cos(1/x) as x increases without bound.

The second is a bit trickier, maybe. As it is written, x does not matter and you have sin(2). I suspect you intended to write sin(2x)/x. One could observe that -1 <= sin(2x) <= 1 and you'ld be about done.

 

[fdragon]

Yeah thanks for the help.... I think its just best to look at the graph and see which y value the graph is going towards or seeing whether or not the graph goes to negative or positive infinity! Does that make sense?

 

[tkhunny]

That is the idea, but you may wish to recall that the early inventors of limits didn't have fancy calculators or computer programs. Thinking about it on a different level will help. In particular, a graph will give you a hint whether it converges to a single value and some clue as to what that value might be. The graph WILL NOT give you a good value without fail. For example, if the actual limit is '2', you are likely to guess that from the graph. If the actual limit is \(\displaystyle \frac{2}{3}\pi\,+\frac{7}{19}\), you are not likely to guess that from any reasonable picture.